the geometry of soft materials: a primer · the geometry of soft materials: a ... an overview of...

REVIEWS OF MODERN PHYSICS, VOLUME 74, OCTOBER 2002

The geometry of soft materials: a primer

Randall D. Kamien*

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia,Pennsylvania 19104

(Published 11 September 2002)

This article presents an overview of the differential geometry of curves and surfaces using examplesfrom soft matter as illustrations. The presentation requires a background only in vector calculus andis otherwise self-contained.

CONTENTS

I. Introduction 953II. Local Theory of Curves 953

A. Conformations of polymers: motivating ageometrical description of curves 953

B. Frenet-Serret apparatus: DNA and other chiralpolymers 954

III. Global Theory of Curves 956A. Fenchel’s theorem: energetic bounds on closed

curves and knots 956B. Mermin-Ho relation: basis vectors to the rescue 957C. Link, twist, and writhe: dynamics of twist-storing

polymers 958IV. Local Theory of Surfaces 961

A. Area element: minimal surfaces 961B. Mean and Gaussian curvature: energetics of

membranes 964V. Global Theory of Surfaces 965

A. Gauss-Bonnet theorem: foams on curvedsurfaces 965

B. Euler characteristic and the genus: defects onsurfaces 967

VI. Three Dimensions and Beyond 969Outlook 970Acknowledgments 970References 970

I. INTRODUCTION

Though geometry is a common part of our earlyschooling, a rigorous and thorough education in physicsusually tries to expunge it from our thought. Because oftheir predictive powers, there is good reason to empha-size numbers and formulas. Analytic geometry is theusual emphasis, while classical geometry is relegated topopular expositions. Differential geometry is a bridgebetween shapes and analytic expressions and is often theappropriate language for modern physics. Nonetheless,when necessary, geometry is often slipped in as a bitteror, at least, tasteless pill—just enough is presented to geton with the analysis. In these lectures I have made anattempt to introduce the basics of differential geometryin the style of a mathematics text: the ideas are groupedby mathematical subject as opposed to physical subject.Nonetheless, I follow each newly developed topic with

*Electronic address: [email protected]

0034-6861/2002/74(4)/953(19)/$35.00 953

an example from soft matter, not only to illustrate theusefulness of the mathematical structure but also to aidthe reader with a concrete example.

This is by no means a textbook and many details ofthe physics are left for the diligent reader to find in thereferences. For technical details of many of the topicsdiscussed here, the reader is referred to Elements of Dif-ferential Geometry by Millman and Parker (Millman andParker, 1977), though any standard reference on classi-cal differential geometry should suffice. A word on no-tation: I have tried throughout to explicitly display thefunctional dependence of all the fields and functions informulas. However, sometimes this would make the no-tation awkward and the dependencies are dropped. Ineach case the context should make clear any lack of pre-cision.

Finally, I have tried to reduce as much as possible theuse of the powerful formalism of differential geometry.While this formalism is useful for performing complexcalculations unambiguously, great expertise is often re-quired to extract the geometrical and physical meaningof these calculations. An excellent complement to thesenotes are the lecture notes by Francois David (David,1989) which present the mathematical elegance and logi-cal compactness of this subject.

II. LOCAL THEORY OF CURVES

A. Conformations of polymers: motivating a geometricaldescription of curves

Random walks abound in physics. They are the basisfor understanding diffusion, heat flow, and polymers.However, polymers are the most interesting of the three:polymers, unlike diffusing particles, leave a ‘‘tail’’ behindthem which they must avoid. They are described by self-avoiding random walks. As an introduction to the powerof geometrical modeling, in this section we will considerthe behavior of stiff polymers at the shortest length scaleamenable to a continuum analysis. At these scales, poly-mers are not random walks at all, but should be thoughtof as stiff rods. We take this as our starting point. Whenwe describe a polymer as stiff, we are ascribing to it anenergy cost for being bent. To model this, we consider acurve, R(s), parametrized by its arc length s and con-struct the tangent vector to our curve T(s)5dR(s)/dsat a point s on the curve. In the next section we willshow that the tangent vector is of unit length. If the

©2002 The American Physical Society

954 Randall D. Kamien: The geometry of soft materials: a primer

tangent vector is constant along the curve then it is astraight line and does not bend. Thus the energy shoulddepend on derivatives of the unit tangent vector T(s).Indeed we call the magnitude dT(s)/ds the curvature ofthe curve. In the next section we will discuss the geom-etry of curves in greater detail. In the meantime, wewrite the energy as

Ecurv512

AE0

LFdT~s !

ds G2

ds , (1)

where s is a parameter that measures the arc length ofthe curve, and A is a measure of the stiffness. To studythe statistical mechanics of the curve we write the parti-tion function for T:

Z~TL!5ET(0)5 z

T(L)5TL@dT#e2Ecurv[T]/kBT. (2)

This is the partition function for the curve which startswith its tangent vector along the z direction and endswith its tangent vector equal to TL . Note that if we cancalculate ^T(s)T(s8)&, then we can integrate with re-spect to s and s8 to obtain

^@R~L !2R~0 !#2&5E0

LdsE

0

Lds8K dR~s !

ds•

dR~s8!

ds L .

(3)

Calculating the correlation function of the tangent vec-tors proves to be straightforward. Though there aremany ways to proceed (Doi and Edwards, 1986), we willchoose here an analogy with quantum mechanics. In Eq.(2) let s5it . Then the partition function becomes

Z~TL ,L !5ET(0)5 z

T(L)5TL@dT#expH i

kBT E0

2iLdt

A

2 S dTdt D 2J ;

(4)

we recognize this as the path-integral solution to Schro-dinger’s equation for a single particle where kBT re-places \ , A replaces the particle’s mass, and T is theposition of the particle. Since T(t) lives on the unitsphere, this is just quantum mechanics on a sphere. TheSchrodinger equation is

2kBT

i

]Z

]L52

~kBT !2

2AL2Z , (5)

where L is the angular momentum operator. Changingback to our original coordinate s and defining Lp[A/kBT we have

]Z~T,s !

]s5

12Lp

L2Z~T,s !. (6)

We will soon discover that Lp has an important interpre-tation.

Since the polymer is the same all along its length, wehave ^T(s)•T(s8)&5^T(s2s8)•T(0)&. Using polar co-ordinates and enforcing the limits of integration in Eq.(4) so that T(0)5 z , we discover that we are interestedin ^cos u(S)& at S5s2s8:

Rev. Mod. Phys., Vol. 74, No. 4, October 2002

^cos u~S !&5

E21

1d~cos u!cos u Z~cos u ,S !

E21

1d~cos u!Z~cos u ,S !

, (7)

where we have been sure to normalize Z to get a prob-ability. The expectation value we seek satisfies a differ-ential equation since Z(cos u,S) satisfies Eq. (6). More-over, since L2 is a Hermitian operator, we have

d^cos u~S !&dS

51

2Lp^L2 cos u~S !&52

1Lp

^cos u~S !&,

(8)

where we have used the fact that L2 cos u522 cos u andL2(1)50. It follows that

^cos@u~s !2u~s8!#&5e2us2s8u/Lp. (9)

Integrating this expression as in Eq. (3), we have

^@R~L !2R~0 !#2&52Lp~L2Lp1Lpe2L/Lp!. (10)

From Eqs. (9) and (10) we glean the meaning of Lp .The first equation shows that the tangent vectors alongthe curve are uncorrelated after a distance Lp . For thisreason Lp is called the persistence length (de Gennes,1970). The second equation shows us that for L muchlonger than Lp , a stiff rod behaves as a random walk:i.e., the average square distance that is traveled is pro-portional to the number of steps or length of the walk,R2}LLp . For L much shorter than Lp , we may expandEq. (10) to see that R2}L2.

The physics of stiff rods can be used to study otherphenomena. For instance, vortices in fluids, superfluids,and superconductors have a bending stiffness arisingfrom self-interactions. In the past decade the physics ofstiff rods has been adapted to study the mechanicalproperties of DNA (Marko and Siggia, 1994, 1955), andhas been augmented to include twisting degrees of free-dom (Kamien et al., 1997; Marko, 1997), the topologicalconstraints imposed by self-avoidance (Vologodskiiet al., 1992) and by closed loops (Moroz and Kamien,1997; Moroz and Nelson, 1997, 1998), and the effects ofsequence disorder (Bensimon et al., 1998; Nelson, 1998).

B. Frenet-Serret apparatus: DNA and other chiralpolymers

In the previous section we have seen that a simplegeometrical description of polymers leads to a precisedescription of their conformational behavior in a varietyof regimes. It is now time to discuss more carefully thegeometry that went into making the expression for en-ergy in Eq. (1). A curve in three dimensions is a vector-valued function R(s)5@x(s),y(s),z(s)# that dependson s , a parameter that runs along the curve. Though wemay choose to label points along the curve as we wish, itis usually most convenient to let s be the arc lengthalong the curve. We will see how this simplifies ourequations shortly. If the curve is L long, then s runs

955Randall D. Kamien: The geometry of soft materials: a primer

from 0 to L . The first thing to do is to construct the unittangent vector to the curve: as we have already asserted,the magnitude of the rate of change of the unit tangentvector is the curvature k of the curve. We have

T~s !5R8~s !

iR8~s !i , (11)

where X8(s) denotes the derivative of X(s) with respectto s , and iXi is the length of the vector X. We havedivided by the length of R8(s) to make T(s) a unit vec-tor. However, if s measures the arc length along thecurve then iR8(s)i51! To see this, recall that the lengthof a curve R(t) from t50 to t5t f can be found by add-ing (integrating) the length of pieces of the curve to-gether, each AdR(t)•dR(t) long:

L~ t f!5E0

tfAdR~ t !•dR~ t !

5E0

tfdtAdR~ t !

dt•

dR~ t !

dt. (12)

If we choose t5s to be the arc length then we have

L5E0

LdsiR8~s !i , (13)

where the upper limit is the same as the length of thecurve. Differentiating both sides of Eq. (13), we see thatiR8(s)i51.

Having constructed the unit tangent vector, we maynow take its derivatives to obtain the curvature. Sincethe derivative of a vector is another vector, we write

T8~s !5k~s !N~s !, (14)

where k(s) is the curvature at s and N(s) is a new vec-tor, the unit normal vector, which is also unit length, sothat iT8(s)i5uk(s)u. It is convention to choose k(s) tobe always positive—the sign can alway be absorbed inthe direction of N(s). Note that T(s)•N(s)50 since thederivative of any unit vector is perpendicular to itself; ifiX(s)i51 then

d

ds@X~s !•X~s !#5

d

ds@1# ,

2X~s !•X8~s !50. (15)

Of course, as we move along s , the normal vectorchanges direction as well. Changes in the direction ofthe normal vector can come from two contributions: thenormal can change by rotating towards or away from thetangent vector (of course the pair rotate together to stayperpendicular). It can also change by rotating aroundthe tangent vector. The former case corresponds to thecurve staying in the same, flat plane, while the secondcorresponds to rotations of the plane in which the curvelies at s . This plane is known as the osculating plane,from the Greek word for kissing. Moreover, since N8(s)is perpendicular to N(s), we must introduce a third unitvector to account for changes in the osculating plane.We choose B(s)5T(s)3N(s) where 3 denotes the


cross product, as shown in Fig. 1. This vector, the binor-mal vector, is a unit vector perpendicular to both T(s)and N(s). We then have

N8~s !5a~s !T~s !1t~s !B~s !, (16)

where a(s) is some function of s and t(s) is called thetorsion of the curve. It is a measure of the rate of changeof the osculating plane. Why do we not give a(s) aname? Because we note that by differentiating the rela-tion T(s)•N(s)50 we get

05T8~s !•N~s !1T~s !•N8~s !5k~s !1a~s !, (17)

so a(s)52k(s) and Eq. (16) becomes

N8~s !52k~s !T~s !1t~s !B~s !. (18)

Finally, we may calculate B8(s) to complete ouranalysis of the curve. We have

B8~s !5T8~s !3N~s !1T~s !3N8~s !

5k~s !N~s !3N~s !2k~s !T~s !3T~s !

1t~s !T~s !3B~s !,

52t~s !N~s ! (19)

where the last line follows from the rule a3(b3c)5b(a•c)2c(a•b). Putting Eqs. (14), (18), and (19) to-gether we have the Frenet-Serret equations for a curvein three-dimensions:

d

ds F T~s !

N~s !

B~s !G5F 0 k~s ! 0

2k~s ! 0 t~s !

0 2t~s ! 0G F T~s !

N~s !

B~s !G . (20)

This shows that k(s) is the rate of rotation of T(s)about B(s) and similarly, t(s) is the rate of rotation ofN(s) about T(s). Written as one matrix equation, theFrenet-Serret formula tells us something very important:given a curvature k(s) and a torsion t(s), we can recon-struct our entire curve up to a translation (since we canchange the origin) and a rotation „since we can rotatethe initial orthonormal triad $T(0),N(0),B(0)%…. Oncewe have set the location and orientation of that triad,however, the entire curve is determined by only two pa-rameters, not three as one might have thought.

There is a difficulty with the Frenet-Serret frame:when the curvature vanishes, N(s) is not well defined

FIG. 1. A curve in space. At R there is a tangent vector T, anormal vector N, and the binormal vector B.


and therefore B(s) and, more importantly, the torsionare not defined either. Thus curves that have straightsegments are problematic from the point of view of theFrenet-Serret frame. Moreover, if we consider a helix

R~s !5F r cosS qs

A~qr !211D ,

r sinS qs

A~qr !211D ,

s

A~qr !211G (21)

then the curvature and torsion are constant:

k~s !5q2r

11~qr !2 ,

t~s !5q

11~qr !2 . (22)

As r→0 we approach a straight line. However, thoughthe curvature vanishes in this limit, the torsion does not!This is a problematic feature of the Frenet-Serretframe—since the torsion is the magnitude of the deriva-tive of N(s), it is only a meaningful quantity whenN(s)5T8(s)/k(s) is unambiguously defined, and thisrequires that k(s)Þ0. In Sec. III.B, we will offer a dif-ferent frame that does not suffer from this problem.

Knowing that there are only two parameters neededto describe a space curve, we can now augment Eq. (1)to include other effects. One interesting effect (Harriset al., 1999) is the behavior of chiral polymers. While thecurvature does not distinguish between left- and right-handed helices, we can see from Eq. (22) that the tor-sion is sensitive to the sign of q . Thus we might addterms to Eq. (1) to favor a particular chirality for thestiff polymer. We might be tempted to write

E5Ecurv1E*

5E0

LdsH A

2k2~s !1

B

2@t~s !2t0#2J . (23)

Though this energy appears acceptable, and favors anaverage torsion t5t0 , it only accentuates the ambiguityof t when k50. Moreover, since Eq. (1) is a functionalof R(s), we should construct the new energy only interms of R(s) and its derivatives. In addition to@R9(s)#25@T8(s)#2, we can also construct the term (Ka-mien et al., 1997; Marko, 1997)

E* 52a

2 E0

LdsT~s !•@T8~s !3T9~s !# . (24)

From the Frenet-Serret formula, we then have

E5E0

LdsH A

2k2~s !1

C

4k4~s !2

a

2k2~s !t~s !

1b

2k2~s !t2~s !J , (25)

where A , C , a, and b are all positive. Note that in thespecial case of constant curvature and torsion, @T(s)


3T9(s)#25k2t2. We have added this term and the extraquartic term to E for reasons that will become clear inthe following. This form does not suffer from the torsionambiguity: when k(s)50 the torsion drops out of Eq.(25). One can minimize this energy for the helix (21),which has constant curvature and torsion, to find theground-state conformation of a chiral polymer. For ap-propriate parameter values, the tendency for torsion canovercome the tendency to be straight and both k and twill be nonvanishing.

III. GLOBAL THEORY OF CURVES

A. Fenchel’s theorem: energetic bounds on closed curvesand knots

When we consider the energetics of closed curves, it isclear that the curvature cannot vanish everywhere (orthe curve will not close on itself). We can actually estab-lish a lower bound on the curvature energy by appealingto a theorem on the total curvature of the curve. Notonly will this be useful when considering closed polymerloops, the technology of the proof will help us later onwhen we consider the geometry of surfaces.

Fenchel’s theorem states that the total curvature of aclosed curve is at least 2p. This is not unreasonable: acircle of radius R has curvature 1/R and so the totalcurvature is 2pR/R52p . In general, other closedcurves only have more curvature (in fact, only planarconvex curves have an integrated curvature of exactly2p). The statement of the theorem is

R0

L

k~s !ds>2p (26)

for any closed curve, where, as usual, s is the arc lengthof the curve. Before we prove this fact, note that via theCauchy-Schwarz inequality we have

R0

L

~1 !k~s !ds<A R0

L

k2~s !dsA R0

L

12ds . (27)

Squaring both sides and using Eq. (26) we find that thecurvature energy (1) satisfies

FIG. 2. The tangent spherical map. The tangent vector of thecurve T traces out a curve G on the unit sphere. The length ofG is the integrated curvature along the original curve.


Ecurv512

kBTLp R0

L

k2~s !ds>2p2kBTLp

L. (28)

To prove Fenchel’s theorem we introduce the tangentspherical image (Millman and Parker, 1977) of the curve:we take the tail of the unit tangent vector T(s) at R(s)and map it to the center of the unit sphere. As shown inFig. 2, the tip of the tangent vector then traces out acurve G on the surface of the unit sphere as s goes from0 to L . The differential of length of the curve on thetangent sphere is iT(s)8ids , in analogy with Eq. (13).But the magnitude of T8 is just the curvature, so we findthat the length of G on the unit sphere is

l5E0

Lk~s !ds , (29)

the total curvature of our original curve R(s). Now wenote that by definition of T(s),

R~s !2R~0 !5E0

sT~s8!ds8, (30)

so if the curve is closed, R(L)5R(0), and the left-handside of Eq. (30) is 0. This is really three equations, onefor each component of T. It tells us that the curve G onthe tangent sphere can never be in only one hemisphere.In order for the curve to close, it must turn around. Butif the tangent spherical map were in one hemisphere, wecould take it to be the upper hemisphere where Tz(s).0. Yet if Tz(s) is always positive then its integral can-not vanish! Therefore, given any hemisphere on the tan-gent sphere, the curve must be in it and its complement.But this means that G must be at least the length of theequator, 2p. We have thus proven Eq. (26). The value ofthis result is not just the bound on the energetics of aclosed polymer. It has introduced us to the tangent map.As we will see in the following it is very useful to takevectors off of curves and surfaces and translate them tothe center of the unit sphere. This was our first taste ofthis procedure.

In closing this section we mention the Fary-Milnortheorem which pertains to the integrated curvature of anon-self-intersecting closed knot. Not surprisingly, thistheorem states that a knot has to go around at leasttwice:

R0

L

k~s !ds>4p . (31)

Using the same reasoning that led to Eq. (28), we have

kBTLp R0

L

k2~s !ds>~4p!2kBTLp

L(32)

for a knotted closed curve.

B. Mermin-Ho relation: basis vectors to the rescue

Up until this point, we have been able to study thegeometry of lines without the introduction of a frame:that is, all our results rely on the original tangent vector


T(s) and its derivatives—at no point were we requiredto choose a basis or coordinate system to calculate anyquantities. Unfortunately, we cannot continue our dis-cussion of lines and we certainly cannot discuss surfaceswithout introducing some more paraphernalia. We needto introduce a set of spatially varying basis vectors inorder to define quantities in addition to the tangent vec-tor, the curvature, and the torsion. We need to makesure, however, that all of our physical quantities do notdepend on our arbitrary choice. In this section we willderive and explain the Mermin-Ho relation (Merminand Ho, 1976), originally derived in the context of super-fluid 3He-A , a phase characterized by an order param-eter with two directions, a ‘‘long’’ direction and a‘‘short’’ direction perpendicular to it. Though this sec-tion is the most technical, it is also straightforward. For-tunately, once we have established this result we will beable to make use of it again and again in the following.

Often we have a vector n(x) defined everywhere on asurface or in space. It could be the director of a liquidcrystal, the normal to a surface or some other field ofinterest. We now are interested in a vector N(x) which isalways perpendicular to n(x). This vector might point tothe nearest neighbor or in some special direction on thesurface. To define this vector, we introduce two new unitvectors, e1(x) and e2(x), which are defined to beperpendicular to n(x) and each other so that$e1(x),e2(x),n(x)% is an orthonormal triad and e1(x)3e2(x)5n(x). A vector which is defined relative to thespatially varying plane defined by e1(x) and e2(x) willchange not only because its true direction changes (rela-tive to a fixed triad) but also because the basis vectorschange. If N(x) is a unit vector then N(x)5cos@u(x)#e1(x)1sin@u(x)#e2(x) is always perpendicu-lar to n. Its derivatives in the i direction are vectors aswell. In the plane normal to n their components are

e1~x!•] iN~x!52sinu~x!@] iu~x!2e1~x!•] ie2~x!# ,(33)

e2~x!•] iN~x!5cos u~x!@] iu~x!1e2~x!•] ie1~x!#

5cos u~x!@] iu~x!2e1~x!•] ie2~x!# , (34)

where we have used the fact that e1(x)•e2(x)50 in thefinal equality. Notice that the derivatives of N(x) dependon both gradients of u(x) as well as derivatives of thespatially varying basis vectors ei(x). Since the basis vec-tors were chosen arbitrarily, one might be concernedthat the derivatives of N(x) are poorly defined. How-ever, there is a concomitant change in u(x) wheneverthe basis vectors change so that the gradients of N(x)are well defined. The problem is with gradients of u(x).

To disentangle the arbitrary dependence on basis vec-tors, we start by considering a vector N0(x) which isconstant in the instantaneous plane perpendicular ton(x) so that Eqs. (33) and (34) both vanish. For N0(x) tobe constant in the plane perpendicular to n(x), u(x)must be equal to some u0(x) with “u0(x)5e1(x)•“e2(x)[V(x), where the last equality defines a newvector field V(x)5e1(x)•“e2(x) called the spin connec-tion. Note that we can only solve this equation for u0(x)


if “3V50. If we now consider an arbitary vector per-pendicular to n(x), N(x), we should focus not on gradi-ents of u(x), but rather on @u(x)2u0(x)# , which is ameasure of how much u(x) deviates from its ‘‘constant’’value. But “u0(x)5V(x), so “@u(x)2u0(x)#5“u(x)2V(x). Thus we see that by subtracting V(x) from“u(x), we remove that part of u(x) that is induced by aspatially varying basis. Moreover, even if “3V is non-vanishing, we can generalize this discussion to form thecombination Du(x)[“u(x)2V(x), the covariant de-rivative. It measures the true changes in u(x), relative tou0(x). If @u(x)2u0(x)# is a smooth field then “

3“@u(x)2u0(x)#50, or

“3“u~x!5“3V~x!. (35)

We might be tempted to set the left-hand side equal tozero, since usually the curl of a gradient vanishes. How-ever, as we are about to show, the curl of V(x) is notalways zero! This means that there must be some sort ofsingularities or defects in u(x) [and u0(x)—the differ-ence u(x)2u0(x) is smooth]. We will talk more aboutdefects in Sec. V.B.

The ith component of the curl is

@“3V~x!# i5e ijk] j@e1a~x!]ke2

a~x!#

5e ijk@] je1a~x!#@]ke2

a~x!#

1e1a~x!e ijk] j]ke2

a~x!

5e ijk@] je1a~x!#@]ke2

a~x!#10, (36)

where the last term vanishes due to the antisymmetry ofe ijk and we have used indices to make the calculationunambiguous.1 We have used the Einstein summationconvention of dropping summation signs for repeatedindices. Unless otherwise indicated, an index should besummed over if it appears twice in any formula. Boththe greek and roman indices run from 1 to 3. Now con-sider the object ] je1

a(x). Since e1(x) is a unit vector, itsderivative is perpendicular to it. Therefore we can write] je1

a(x) in terms of the basis vectors na(x) and e2a(x):

] je1a~x!5Aj~x!na~x!1Bj~x!e2

a~x!. (37)

Similarly, we can do the same for ]ke2a(x):

]ke2a~x!5Ck~x!na~x!1Dk~x!e1

a~x!. (38)

Putting these expressions into Eq. (36) and using theorthogonality of our triad, we find

@“3V~x!# i5e ijkAj~x!Ck~x!. (39)

By taking the dot product of Eq. (37) with n(x), we findAj(x)5nb(x)] je1

b(x)52e1b(x)] jn

b(x). Similarly, Ck(x)52e2

g(x)]kng(x) and so

1The only tensor on which we will rely is the antisymmetrictensor e ijk . It is defined by

eijk5H11 if ijk is an even permutation of 123

21 if ijk is an odd permutation of 123

0 if any two of i , j , or k are the same.


@“3V~x!# i5e1b~x!e2

g~x!e ijk@] jnb~x!#@]kng~x!# .

(40)

We see then that the curl of V(x) does not always van-ish, but it appears that it depends on our arbitrary vec-tors e1(x) and e2(x). Note, however, that if we inter-change the indices b and g in Eq. (40) that it isequivalent to interchanging j and k , which would intro-duce a minus sign. Thus we have

@“3V~x!# i512

@e1b~x!e2

g~x!2e1g~x!e2

b~x!#

3e ijk@] jnb~x!#@]kng~x!#

512

eabgna~x!e ijk] jnb~x!]kng~x!, (41)

where we have used the orthonormality of$e1(x),e2(x),n(x)%. This is the celebrated Mermin-Horelation (Mermin and Ho, 1976). Note that because ofthe antisymmetry of e ijk , it is unnecessary to keep thebrackets around the gradients of n(x).

This relation between n(x) and vectors perpendicularto n(x) is rather remarkable. Note that our choice ofe1(x) and e2(x) is arbitrary so that V(x) is not a con-stant of the geometry. However, Eq. (41) shows that“3V(x) only depends on n(x), and not our choice ofbasis vectors! This is why the Mermin-Ho relation hasbeen introduced in this section on global properties (Ka-mien, 1998): if we have a closed curve G of length L ,then

RG@“u~x!2V~x!#•dR

5@u~L !2u~0 !#2EEM

@“3V~x!#•dS

5@u~L !2u~0 !#

2EEM

12

emnreabgna~x!]nnb~x!]rng~x!dSm , (42)

where we have used Stokes theorem to change an inte-gral around a curve G into an integral over a cappingsurface M . In Eq. (42) dS is an element of area. Thus ifwe consider changes in a vector around a closed curve,our choice of ei(x) is irrelevant. Though it is necessarythat N(L)5N(0), u(L)2u(0) can change by an inte-gral multiple of 2p. The rest of the change in the direc-tion of N(x) comes from the geometry of n(x). We willsee how this extra term plays an important role in thenext section, in the physics of surfaces and the physics ofdefects in three dimensions.

C. Link, twist, and writhe: dynamics of twist-storingpolymers

One of the more interesting stiff polymers is DNA.Though it is well known that it has great biological sig-nificance (Watson and Crick, 1953), it is of interest inmaterials for at least two other reasons. First, the persis-


tence length of DNA is quite long, roughly 50 nm. Moreimportantly, because there are a plethora of enzymesavailable to cleave DNA, a sample of monodispersepolymers can be prepared much more readily than inany synthetic system. We have already discussed confor-mations of chiral polymers like DNA in Sec. II. How-ever, DNA has one more interesting element: it canform into a closed loop. This is interesting because DNAis actually a double stranded helix. Therefore the num-ber of times that one strand wraps around the other isfixed when the loop is closed (when the strand is openthe helix can unravel). Thus there is a conserved quan-tity and this leads to a constraint on the possible dynam-ics of the double strand. More generally, any polymersthat cannot unwind along their axis are known as twist-storing polymers.

When we have two closed curves G and G8, we canassign a linking number Lk to them which counts thenumber of times one loop passes through the other.There is a simple way to calculate this given the twocurves R(s) and R8(s) using an analogy with Ampere’slaw. If we think of the first curve G as being a wire car-rying a current I , then we know that the line integral ofthe generated magnetic field B around the closed curveG8 is 4pnI , where n is the number of times that thecurrent passes through the closed loop.2 Setting I51, wehave

Lk5n51

4p RG8

B~x8!•dx8. (43)

We may calculate the resulting field B(x8) from the wireby use of the Biot-Savart Law:

B~x8!5 RG

dl3 rr2 5 R

G

dx3@x82x#

ux82xu3 . (44)

Putting these together we find that (with the curves hav-ing length L and L8, respectively)

Lk51

4p RGR

G8

@x2x8#•~dx3dx8!

ux2x8u3

51

4p R0

L

ds R0

L8ds8FdR~s !

ds3

dR8~s8!

ds G•

@R~s !2R8~s8!#

uR~s !2R8~s8!u3

[G~G ,G8!, (45)

the last equality defines G(G ,G8), the Gauss invariant.Thus two closed curves that cannot separate and rejoinmust keep this invariant constant.

While this may be elegant, it is not especially usefulwhen studying molecules like DNA. At long lengthscales, DNA appears as a single filament. It would beuseful to recast the linking number in terms of the singlepolymer picture of DNA. To do this, we consider two

2For simplicity we work in cgs units with c51.


curves that are close together (White, 1969; Fuller,1971). The first curve is R(s), while the second curve is

R8~s8!5R~s8!1eu~s8!, (46)

where, as is the tradition, e is a small number and u(s8)is a unit vector that is perpendicular to the tangent vec-tor dR(s8)/ds5T(s8). Inserting these expressions intoEq. (45), we have

Lk51

4p R0

L

ds R0

L8ds8 T~s !

3FT~s8!1edu~s8!

ds G• @R~s !2R~s8!2eu~s8!#

uR~s !2R~s8!2eu~s8!u3 .

(47)We now want to take e to zero. While the numerator inEq. (47) does not make this a problem, note that thedenominator vanishes whenever s5s8. Thus as long asus2s8u>d we can take e→0 (Kleinert, 1990). We thushave

Lk51

4p R0

L8ds8H E

0

s82dds1E

s81d

LdsJ

3T~s !3T~s8!•@R~s !2R~s8!#

uR~s !2R~s8!u3

11

4p R0

L8ds8E

s82d

s81dds T~s !

3FT~s8!1edu~s8!

ds G• @R~s !2R~s8!2eu~s8!#

uR~s !2R~s8!2eu~s8!u3 .

(48)

The first integral bears a resemblance to the Gauss in-variant, while the second integral depends on the vectoru(s). Since d is small, we can expand R(s) around s8 tofind

R~s !5R~s8!1~s2s8!T~s8!1¯ (49)

and

T~s !5T~s8!1~s2s8!dT~s8!

ds1¯ . (50)

Inserting this into the second integrand and using thefact that T(s)•u(s)50, we find

FIG. 3. The tangent spherical map. The area of the patch Mon the sphere is, through Stokes theorem, the line integralaround G of e1•¹e2 .


Lk51

4p R0

L

ds R0

L8ds8 T~s !3T~s8!•

R~s !2R~s8!

uR~s !2R~s8!u321

4p R0

L8ds8E

s82d

s81dds

3

FT~s8!1~s2s8!dT~s8!

ds G3FT~s8!1edu~s8!

ds G•eu~s8!

@~s2s8!21e2#3/2 , (51)

where we have taken e→0 in the first integral and haveused T(s8)3T(s8)50 in the second. Note that the sdependence of the second integral is manifest and wemay do the integration over s , let e go to zero first andthen let d go to zero. We have separated the link intotwo integrals. The first is called the writhe, Wr , and itonly depends on the curve R(s). The second is calledthe twist, Tw , and it depends on u(s). We have

Wr51

4p R0

L

ds R0

L

ds8 T~s !3T~s8!•R~s !2R~s8!

uR~s !2R~s8!u3 ,

(52)

Tw51

2p R0

L

ds T~s !•Fu~s !3du~s !

ds G (53)

and thus we arrive at White and Fuller’s celebrated re-sult (White, 1969; Fuller, 1971):

Lk5Tw1Wr . (54)

Though the expression for writhe (52) bares a strongresemblance to that for the Gauss invariant (45), it is notthe same. In the Gauss invariant we were consideringtwo different curves that did not touch and there was noneed to expunge a singularity from the integration. Theexpression for writhe, on the other hand, is for the samecurve and is only defined in the limit described above.Note further that the writhe is nonlocal, while the twistis local. This is the price we must pay: the link is topo-logical, the writhe depends only on the backbone R(s)but is nonlocal, and the twist is local but we must knowabout u(s), i.e., the other curve.

Note that the expression for the twist (53) suggests ause of the results of Sec. III.B. Since u(s) is perpendicu-lar to T(s), we may expand it in a set of basis vectorse1(s) and e2(s) both perpendicular to T(s), so thatu(s)5cos u(s) e1(s)1sin u(s) e2(s). Then

T~s !•Fu~s !3du~s !

ds G5]su~s !2e1~s !•]se2~s !, (55)

where we have used the fact that e13e25T (plus cyclicpermutations). But this is the form we considered in thediscussion of the change in a vector. Following the argu-ment at the end of the previous section, we find that

Tw51

2p R0

L

ds @]su~s !2e1~s !•]se2~s !#

5m2EEM

14p

emnreabgTa~x!]nTb~x!]rTg~x!dSm

5Lk2Wr , (56)


where m is an integer [since u(L)2u(0)52pm]. Wewould be tempted to identify the second integral Wr asthe writhe, since we have an equation that reads just asEq. (54). This is not always correct and depends on ourchoice of basis vectors. However, it does establish aweaker result of Fuller’s (Fuller, 1978) that the writhe isthe last term in Eq. (56) mod 1. As we will see in thenext section, the integral has a geometric meaning: it isthe area swept out by the tangent curve on the tangentspherical map, as shown in Fig. 3.

The difficulty with writhe is that it is nonlocal andtherefore cannot be easily included in a local set of dy-namical laws—the conformation of the entire curvemust be known to calculate writhe. Moreover, writhe isonly defined for closed curves and so the identity (54)would not appear to apply to open strands. While thetotal amount of writhe including the integral part is im-portant for calculating ground states of a particulartwisted ribbon (Julicher, 1994; Marko and Siggia, 1994;Fain et al., 1996; Fain and Rudnick, 1999) when consid-ering changes in writhe, the constant integer part is lessimportant. If the time scale for the diffusion of twistalong the polymer is long enough, then even an openstrand should feel the constraint of conserved link.

To this end, a useful result for the change in writhe ofa curve as a function of time t (Aldinger et al., 1995),

] tWr~ t !51

2p R ds T~s ,t !•@] tT~s ,t !3]sT~s ,t !# , (57)

can be used. Note that this result follows from the pre-ceding discussion: if we choose a coordinate system,then (57) (multiplied by dt) is the differential of areaswept out by the tangents of two closed curves with tan-gent vectors T(s ,0) and T(s ,dt). Since the twist is reallythe local torsional strain of the polymer, we denote thetwist density as v(s ,t) and then we have

] tLk5 R ds $] tv~s ,t !1] tT~s ,t !•@]sT~s ,t !

3T~s ,t !#%1] tn , (58)

where n is the integer difference between Wr and Wr .Since continuous evolution cannot lead to discontinuouschanges in the integer n and since link is conserved wehave

05 R ds$] tv1] tT•@]sT3T#%. (59)

This conservation law need not be satisfied locally: thecurve can twist in one place and writhe at some distant


location to satisfy Eq. (59). In addition, the integer canchange if the curve develops cusps and evolves in a non-smooth way. Because physics is local, however, we mightexpect that linking number is locally conserved andchanges via a ‘‘link current’’ j . This would lead to a localconservation law

]sj5] tv1] tT•@]sT3T# , (60)

which satisfies Eq. (59). This conservation law enforcestotal link conservation since rds]sj[0, yet allows forlocal deviations in the twist and writhe (Kamien, 1998).

This geometrically inspired conservation law has beenverified in numerical experiments (Goldstein et al., 1998;Wolgemuth et al., 2000) which show that there are twomodes of relaxation in the dissipative dynamics: twirling,in which the link relaxes through torsional modes alongthe polymer (like a speedometer cable), and whirling, inwhich the link relaxes through crank-shaft-like motionsof the entire chain.

IV. LOCAL THEORY OF SURFACES

A. Area element: minimal surfaces

While it is easy to generalize the equation of a curveR(s) to an equation for a surface X(s1 ,s2), it turns outthat the theory of surfaces is much richer than that forcurves. In the first place, there is no way to talk aboutarc length when defining s1 and s2 , so we will take themto be arbitrary coordinates of the surface. We can, how-ever, consider the area of a small patch of our surface.To do this, we need to construct vectors tangent to thesurface. As shown in Fig. 4, if we have two vectorsa(s1 ,s2) and b(s1 ,s2) in the surface, then they span asurface area DS5ua(s1 ,s2)u ub(s1 ,s2)usin u, where u isthe angle between the two vectors. We recognize thisexpression for DS as the magnitude of the cross producta(s1 ,s2)3b(s1 ,s2). Since both of these vectors are tan-


gent to the surface, their cross product is parallel to theunit layer normal n(s1 ,s2). Thus we have

DS5un~s1 ,s2!•@a~s1 ,s2!3b~s1 ,s2!#u. (61)

But, with this construction, we can also construct theunit layer normal out of a(s1 ,s2) and b(s1 ,s2):

n~s1 ,s2!56a~s1 ,s2!3b~s1 ,s2!

ia~s1 ,s2!3b~s1 ,s2!i . (62)

Putting this expression together with Eq. (61), we have

DS5@a~s1 ,s2!3b~s1 ,s2!#•@a~s1 ,s2!3b~s1 ,s2!#

ia~s1 ,s2!3b~s1 ,s2!i

5A@a~s1 ,s2!3b~s1 ,s2!#2

5Aa2~s1 ,s2!b2~s1 ,s2!2@a~s1 ,s2!•b~s1 ,s2!#2.

(63)

We now need to construct two, nonparallel vectorsa(s1 ,s2) and b(s1 ,s2). The surface provides uswith two: a(s1 ,s2)5]s1

X(s1 ,s2)ds1 and b(s1 ,s2)5]s2

X(s1 ,s2)ds2 . This tells us that the differential areaelement is

FIG. 4. The magnitude of the cross product of two vectors isthe area of the parallelogram swept out by them.

dS5A@]s1X~s1 ,s2!#2@]s2

X~s1 ,s2!#22@]s1X~s1 ,s2!•]s2

X~s1 ,s2!#2ds1ds2 . (64)

Often we also need the vector surface element dS5n(s1 ,s2)dS , for instance when calculating electric andmagnetic flux. The area of the whole surface M is simply

A5EEMA~]s1

X!2~]s2X!22~]s1

X•]s2X!2ds1ds2 .

(65)

What happens if we choose to parametrize the surface interms of a new set of coordinates? If we have two newcoordinates s1 and s2 defined by s i5s i(s1 ,s2), then bythe chain rule

]siX5

]X]si

5]X]s j

]s j

]si5

]s j

]si]s j

X, (66)

where, as usual, there is an implicit sum over j . Becauseof this sum, it appears that it would be somewhat tediousto reexpress Eq. (65) in terms of the new coordinates.However, we note that if we define a matrix g

g~s1 ,s2!5F ]s1X•]s1

X ]s1X•]s2

X

]s2X•]s1

X ]s2X•]s2

XG , (67)

then the expression in the radical of Eq. (65) is simplythe determinant of this matrix. In component form, thismatrix is gij(s1 ,s2)5]si

X(s1 ,s2)•]sjX(s1 ,s2). This ma-

trix is known as the metric tensor which some peoplefind useful (Minser et al., 1973). Moreover, we note thatthe transformation (66) amounts to matrix multiplica-tion of g(s1 ,s2):


gij~s1 ,s2!5]sk

]si

]sm

]sjgkm~s1 ,s2!, (68)

where gkm(s1 ,s2) is the corresponding matrix in thenew coordinate system. If we define a new matrix O byOij5]s j /]si , then g5OTgO. Thus when we change co-ordinates, we have

A5EEM

Adet g ds1ds2

5EEM

Adet OTgO ds1ds2

5EEM

Adet gudet Ouds1ds2 . (69)

But the determinant of O is just the Jacobian determi-nant of the transformation from si to s i , and soudet Ou ds1ds25ds1ds2 or

A5EEM

Adet g ds1ds25EEM

Adetg ds1ds2 , (70)

and so the area is invariant under coordinate transfor-mations. This invariance (known as diffeomorphism co-variance by the cognoscenti) is useful as it allows us tochoose the most convenient set of coordinates for agiven problem. One often used choice is the so-calledMonge gauge or height representation, where s1 and s2are chosen to be the x and y components of the surfacevector X(s1 ,s2) and the z component is a function of xand y :

x5X1~s1 ,s2!5s1 ,

y5X2~s1 ,s2!5s2 ,

z5X3~s1 ,s2!5h~s1 ,s2!5h~x ,y !. (71)

It is straightforward to check that the expression for thearea (65) becomes the more familiar expression:

A5EEMA11S ]h

]xD 2

1S ]h

]yD 2

dxdy . (72)

The height representation requires that for every x andy , there is only one height or, in other words, the surfacecan have no overlaps. Sometimes it is necessary to breakthe surface up into different regions or patches in orderto make a good height representation.

The height representation proves useful when consid-ering the energetics of fluid membranes. By fluid wemean that the membrane has no internal structure andthe molecules in it can flow freely. Often we are inter-ested in the shape and dynamics of these membraneswhen they are under a uniform applied tension. Soapfilms are a common example, though in many cases theinterface between two fluids or two distinct phases canalso be thought of as a membrane under tension. Thefree energy of this system is simply F5§A , where § isthe surface tension and A is the area of the interface.Minimizing the free energy thus amounts to minimizingthe area of the surface. The expression for the area in


Eq. (72) looks like an action from classical mechanics. Ifwe define the Lagrange density L,

L5A11S ]h

]xD 2

1S ]h

]yD 2

, (73)

then the Euler-Lagrange equation is (Goldstein, 1980)

]

]x

]L]~]xh !

1]

]y

]L]~]yh !

5]L]h

, (74)

or, by inserting Eq. (73),

]

]x S ]xh

A11~“h !2D 1]

]y S ]yh

A11~“h !2D 50. (75)

This equation has an important interpretation in termsof the surface normal. If we construct the two tangentvectors to the surface as we did above, we have

u15]s1X~s1 ,s2!5@1,0,]xh# ,

u25]s2X~s1 ,s2!5@0,1,]yh# , (76)

and so the surface normal is parallel to u13u25@2]xh ,2]yh ,1# . Normalizing this vector gives usn(s1 ,s2)52@]xh ,]yh ,21#/A11(“h)2 and so Eq. (75)becomes “•n50. In other words, a surface which ex-tremizes its area has a divergence-free unit normal. Wewill address this further in the following section. Such asurface is called a minimal surface. Typically, Eq. (75) iswritten equivalently as (Nitsche, 1989) the minimal sur-face equation:

@11~]yh !2#]x2h22]xh]yh]x]yh1@11~]xh !2#]y

2h50.(77)

Though we were seeking a surface with a minimumarea, the Euler-Lagrange equations only guarantee anextremum—we might have a saddle point or, worse, amaximum! Rigorous proofs that the area is minimized ingeneral are difficult. However, we can prove a restrictedresult easily. If the projection of the boundary ]M ontothe xy plane is convex, then it is clear that any over-hangs on the surface add extra area and that the surfaceneed not ‘‘stick out’’ past the boundary of M projectedonto the xy plane. Thus we can use the height represen-tation and solve Eq. (77) subject to a boundary condi-tion h(x ,y)5f(x ,y) on the boundary in the xy plane.As depicted in Fig. 5, we consider an arbitrary surface Mwith the same boundary as the surface of which we areinterested. Since the same range of x and y map outboth M and M , we can define n(x ,y) on M and in thevolume between the surfaces. Define

T5EEM

n~x!•dS, (78)

where n(x) is the unit normal of the original surface M ,and dS is the vector element of surface area, pointingalong the normal to the surface n(x). If we consider thesame integral for the original surface M , then since n(x)


is parallel to dS, we have n(x)•dS5dS the element ofsurface area! Integrating this over the whole surface justgives us the area A . Thus

A5EEM

dS5EEM

n~x!•dS. (79)

Moreover, we have the inequality n(x)•n(x)<1, sincethe dot product of two unit vectors is the cosine of someangle and is therefore less than or equal to 1. We arriveat

T5EEM

n~x!•dS5EEM

n~x!•n~x!dS<EEM

dS5A ,

(80)

where A is the area of the comparison surface. We arealmost done. Note that if we construct a closed surfaceM by gluing together M and M along their commonboundary, then the integral of n(x) over the whole sur-face is

EEM

n~x!•dS5EEM

n~x!•dS2EEM

n~x!•dS5A2T ,

(81)

where the relative minus sign arises because we want thenormal to the closed surface to always point outward.But by Gauss’s law, the first integral in Eq. (81) can beconverted into an integral over the volume V enclosedby M , so

A2T5EEM

n~x!•dS5EEEV“•n~x!dV50, (82)

since M extremizes the area so “•n(x)50. Using theinequality (80) we have A5T<A and thus the compari-son surface has an area at least as large as the original

FIG. 5. The original minimal surface M and the comparisonsurface M.

FIG. 6. Edge (left) and screw (right) dislocations in a smectic-A liquid crystal.


surface M . Since M was arbitrary we have shown thatour surface M minimizes the area.3

Examples of minimal surfaces abound. The smectic-Aliquid crystal phase is a one-dimensional crystal, com-posed of periodic layers of spacing d . Each layer is flu-idlike, with no internal ordering. Thus, in some sense,each layer behaves as a minimal surface and the layerspacing is set by a compression modulus (Kamien andLubensky, 1999). The smectic phase can have topologi-cal defects of two types. As shown in Fig. 6, an edgedislocation adds or ends a layer in the bulk, while ascrew dislocation connects adjacent layers together. Ascrew defect can be represented by the height function

h~x ,y !5nd

2ptan21S y

x D , (83)

where n is an integer. By choosing n to be an integer, weare guaranteed that when going around the origin oncewe move up exactly n layers. It is straightforward tocheck that the screw dislocation is a minimal surface bydirect calculation of Eq. (77).

Another nice example also arises in smectic liquidcrystals. Scherk’s first surface (shown in Fig. 7) (Scherk,1835) is a minimal surface which smoothly connects twosmectic regions which are rotated with respect to eachother. It is composed of an infinite set of parallel, screw-like dislocations (Gido et al., 1993; Kamien, 2001) whichmust be slightly squashed to make the surface satisfy Eq.(77). For those who are interested, the height function isdefined for arbitrary rotation angle a:

h@x ,y ;a#[2secS 12

a D tan21H tanhF12

x sin~a!GtanFy sinS 1

2a D G J .

(84)

It is interesting to note that as a→0 the height functionbecomes that of a simple screw dislocation (83) withnd52p .

3In this context, the vector field n(x) is called a calibration.The interested (or disinterested) reader will find this and otherresults in Morgan (2000).

FIG. 7. Scherk’s first surface. This minimal surface connectstogether two layered structures with different orientations.


B. Mean and Gaussian curvature: energetics ofmembranes

When we discussed curves in three dimensions therewere not many options for the definition of curvature:there was only one coordinate on the curve, and if wewanted something that was independent of our param-etrization, we need only take derivatives with respect toarc length. As we have seen in the last section, however,surfaces can be reparametrized and have many more de-grees of freedom. As a result, there is more than oneway to define curvature, in fact there are two.

We have already hinted at one definition in the discus-sion of minimal surfaces. For curves, we know that astraight line minimizes the distance between two pointsin space, or in other words, k(s)50 is the equation of alength-minimizing curve. Generalizing this to surfaces,we might define one curvature which is 0 for surfacesthat minimize the area spanned by a given boundary.Indeed, this is called the mean curvature, H and H52 1

2 ¹•n(x) where n(x) is the surface normal. Theother type of curvature is the Gaussian curvature K ,which we will define in the following.

In order to understand this better and to introduce K ,we need to step back and consider our surface X(s1 ,s2).Since the curvature of a line is k(s)52T(s)•N8(s), wewould like to take derivatives of the surface normalalong tangent directions of the surface. As in Sec. III.B,we now reintroduce the basis vectors perpendicular ton(s1 ,s2), e1(s1 ,s2), and e2(s1 ,s2), of unit length andmutually orthogonal. We can take directionalderivatives4 of n(s1 ,s2) along ei(s1 ,s2), (ei•“). Nowinstead of one number, we have four, which form a ma-trix:

L52Fe1•@e1•“#n e2•@e1•“#n

e1•@e2•“#n e2•@e2•“#nG . (85)

The two dot products in the matrix are somewhat con-fusing. To be concrete we write this matrix using its in-dices:

Lij52eia~s1 ,s2!ej

b~s1 ,s2!]na~s1 ,s2!

]Xb~s1 ,s2!. (86)

This matrix is also a tensor, known as the Weingartenmap or second fundamental form.5 We can diagonalizethis matrix via a similarity transform L5S21LS. In thisdiagonal basis the entries of L are precisely what we areafter: the first entry in the upper left is the derivative ofthe surface normal along a direction e1 . Moreover, sincethe upper right entry vanishes, this derivative has no

4It may seem odd to use the gradient in these definitions sincen(s1 ,s2) is a function of s1 and s2 . The derivatives of n(s1 ,s2)along the surface tangent vectors ]sj

X(s1 ,s2) are just]sj

n(s1 ,s2). By writing ei(s1 ,s2) as the linear combination ei

5Aij]sjX, the chain rule implies that (ei•“)5Aij]sj

.5The metric tensor g is sometimes called the first fundamental

form.


components along e2 . Therefore the upper left corner isthe curvature of the curve in the surface at (s1 ,s2), tan-gent to e1 . Similarly, the lower right entry of L gives usanother curvature. Note that the two directions e1 ande2 remain orthogonal.6 They are known as the principaldirections on the surface and their associated curvaturesk1 and k2 are the principal curvatures (see Fig. 8).Equivalently, we can define the two principal radii ofcurvature through Ri51/k i . We can easily extract thesecurvatures from the original matrix L. Note that k1k25det L and k11k25Tr L. Moreover, since the trace iscyclic and det AB5det A det B, we see that

Tr L5k1~s1 ,s2!1k2~s1 ,s2!,

det L5k1~s1 ,s2!k2~s1 ,s2!. (87)

The product of the curvatures is known as the Gaussiancurvature, K5k1k2 , while the average of the curvaturesis the mean curvature, H5 1

2 @k11k2# . The Gaussianand mean curvature contain all the information to de-scribe the bending of our surface.

The reader may be concerned that our definitions ofcurvature remain dependent on our choice of basis vec-tors. This is not the case. Indeed, using the expression(86) we have

H5212 (

i51

2

eiaei

b ]na

]xb ,

K5~e1ae1

be2ge2

d2e1ae2

be2ge1

d!]na

]xb

]ng

]xd . (88)

These expressions still look dependent on ei . However,since n is a unit vector, nanb]bna50 and so we have

H5212

@e1ae1

b1e2ae2

b1nanb#]na

]xb

5212

dab]na

]xb 5212“•n, (89)

where we have used the fact that $e1 ,e2 ,n% form an or-

6The orthogonality of the two vectors e1 and e2 follows fromthe fact the S215ST. In other words, L can be diagonalized viaan orthogonal transformation since it is symmetric: L122L21

52(e1ae2

b2e2ae1

b)]bna52n•@“3n#50 when n is normal to asurface, i.e., n5“f/u“fu.

FIG. 8. Curves on a surface (heavy lines) with tangents alongthe principal directions at the point of intersection.


thonormal triad. Thus, as we hinted at the beginning ofthis section, the mean curvature is proportional to thedivergence of the surface normal. Note that the sign ofH depends on the choice of the sign of n.

We have to do a little more work on the expressionfor the Gaussian curvature. The form in Eq. (88) re-minds us of the expressions in the derivation of theMermin-Ho relation. Using the orthonormality of thebasis vectors,

K5e1ae2

g~e1be2

d2e2be1

d!]bna]d ng

5e1ae2

gebdrnr]bna]d ng. (90)

Finally, as in the discussion of Eq. (40), we have

K512

eabgnge ijknk] ina] jn

b5n•@“3V# . (91)

Moreover, the Gaussian curvature has a simple interpre-tation. Consider the normal spherical map defined byanalogy with the tangent spherical map. For each pointon the surface we identify a point on the unit spherewhich corresponds to the surface normal at that point.This map is also known as the Gauss map. Since thesurface element on our surface is just dSm5nmdS , wesee from the discussion at the beginning of Sec. IV.A,that the Gaussian curvature is just the area swept out bythe surface normal on the Gauss map as we move alongthe surface. In other words, if we have a small region Mof our surface, then

EEM

KdS5EEM

12

eabg nge ijk] ina] jn

bdSk . (92)

Thus the Gaussian curvature is the ratio between theinfinitesimal area swept out on the Gauss map and theinfinitesimal area of the original surface to which it cor-responds.

We have seen that the two principal curvatures de-pend only on the surface normal and not on our choiceof coordinates or basis vectors. This is useful when weconsider the energetics of fluid membranes. They arecalled ‘‘fluid’’ because they have no internal structure.Therefore it would be unphysical to build an energy outof anything but the two invariants H and K . Note that ifwe have an open surface there is no distinction betweeninside and outside, so the layer normal n(s1 ,s2) is de-fined only up to a sign. Though the Gaussian curvatureis independent of this sign, the mean curvature is not. Toget around this, we insist that the free energy be even inpowers of H (Canham, 1970; Helfrich, 1973):

FCH5EE dS$2kH21kK%, (93)

where k and k are (confusingly) the standard symbolsfor the bending moduli. This is known as the Canham-Helfrich free energy for fluid membranes. Note that theyboth have units of energy, since the dimensions of K andH2 cancel the dimensions of the surface. Also note thatthe integration is done with respect to the actual surfacearea, so that dS5Adet gds1ds2 . We will see in the fol-


lowing sections that the integral of the Gaussian curva-ture is a constant and so the term proportional to k isusually neglected. We will also discover that not only isthe Gaussian curvature independent of our choice of ba-sis vectors, but that it also can be measured with noknowledge of the surface normal n(s1 ,s2). Finally, if themembrane is tethered then it has internal elastic degreesof freedom. These degrees of freedom couple to the ge-ometry and produce a variety of singular structures (Di-Donna et al., 2002).

V. GLOBAL THEORY OF SURFACES

A. Gauss-Bonnet theorem: foams on curved surfaces

We have seen that there are two sorts of curvaturethat we can consider on a surface, the mean and Gauss-ian curvatures. We have studied both of these by consid-ering deviations of the normal vector to the surface.However, a large part of differential geometry focuseson intrinsic properties, those quantities which can bemeasured without reference to the space in which themanifold is embedded. It turns out that the Gaussiancurvature is intrinsic: by measuring the lengths and di-ameters of small circles entirely in the surface, one candetermine K .

We have all the technology necessary to demonstratethis remarkable property. To show this, we consider apatch on our surface M , with boundary ]M . As in Sec.III.C, we can take the unit normal at each point and mapit to the unit sphere. As shown in Fig. 9, when wetraverse the patch M on the surface, we traverse a patchM on the unit sphere. As we discussed, the area of M isthe integral of the Gaussian curvature of the patch M :

FIG. 9. Geometry for the Gauss-Bonnet theorem. Note thatnow we are using the Gauss map which maps the surface nor-mal n(x) unto the unit sphere.


A5EEM

dS5EEM

KdS

5EEM

12

emnreabgna~x!]nnb~x!]rng~x!dSm , (94)

where we have used Eq. (91). Now consider the bound-ary curve ]M . This curve has a tangent vector T(s) inspace. We know from our discussion of curves thatT8(s)5k(s)N(s). However, while T(s) is also tangentto the sphere, N(s) is, in general, not. The normal to thecurve, N(s), can have a component along the normal tothe surface n(s) and perpendicular to it. If we lived inthe surface and could only make measurements in thesurface, like some kind of bug, then we could only mea-sure the component of the curve’s normal in the surface,the surface normal g(s):

g~s !5T8~s !2@T8~s !•n~s !#n~s !

k~s !A12@N~s !•n~s !#2. (95)

This is a unit vector which lies in the plane tangent tothe surface and which is perpendicular to T(s), that is,the tangent to the boundary ]M of the patch M . Wewould like to define a curvature which measures the ex-tra curvature in the curve, not arising from the curvatureof the surface in which it is embedded. By analogy to theFrenet-Serret formulas, we define the geodesiccurvature7 kg to be

kg~s !5T8~s !•g~s !5k~s !A12@N~s !•n~s !#2. (96)

Since $T(s),g(s),n(s)% form an orthonormal triad, wehave

kg~s !5T8~s !•@n~s !3T~s !#5n~s !•@T~s !3T8~s !# ,(97)

a form we have seen before in the context of theMermin-Ho relation! Since the unit tangent vector liesin the plane perpendicular to n(x), we may write it interms of our arbitrary basis e1(x) and e2(x), introducedin Sec. III.B, T(s)5cos u e11sin u e2 . Using Eq. (97) tocalculate kg , we find that

kg~s !5]su~s !2e1a~s !]se2

a~s !, (98)

where we have used the fact that for a unit vector u(s),]s@u(s)#252u(s)•]su(s)50. In analogy with Eq. (42),we can integrate the geodesic curvature around theboundary of M to find

R]M

kg~s !ds5 R]M

@“u~x!2V~x!#•dR. (99)

As with our discussion of the relation between link,twist, and writhe, we may transform the line integral ofV(x) into a surface integral over M so that

7We may also define the normal curvature, kn(s) which is thecurvature of our embedded curve that is imposed by the sur-face. We define kn(s)5T8(s)•n(s) and thus the total curva-

ture satisfies k(s)5Akn2(s)1kg

2(s).


R]M

]su~s !ds5 R]M

kg~s !ds1EEM

KdS , (100)

where we have used Eq. (41) to rewrite the final inte-grand as the Gaussian curvature (94). Finally, if theboundary curve does not intersect itself, then the tan-gent vector rotates around n(x) exactly once, so u(s)changes by 2p around the curve. We have thus estab-lished the Gauss-Bonnet theorem:

EEM

Kds1 R]M

kgds52p . (101)

By integrating the geodesic curvature around a closedloop, we can calculate the integrated Gaussian curvaturethat we surround. As the loop shrinks ever smalleraround a point x0 , we can calculate the Gaussian curva-ture at that point through division by the surface areaenclosed. This is remarkable since the geodesic curvaturecan be measured without the use of the layer normal! Inother words, since the geodesic curvature is intrinsic, sois the Gaussian curvature! We can take this result a stepfurther by considering regions that have discrete anglesin their boundaries (i.e., polygons). When there is asharp bend in the boundary curve ]M , we cannot calcu-late the geodesic curvature kg(s). We can, however, in-tegrate the geodesic curvature along the smooth parts ofthe boundary. If there are j sharp bends with angles Du j ,respectively, then when we integrate around the bound-ary as in Eq. (99) we will have a deficit of ( jDu j , or, inother words, the smooth part of u(s) does not have tochange by 2p for the curve to come around. We thushave

EEM

KdS1 R]M

kgds1(j

Du j52p , (102)

where we understand that the integral around theboundary should be broken into smooth segments of theboundary. The jump angles account for the discontinui-ties. Note that we can have lines for which kg(s)50.These are the ‘‘straight’’ lines on the surface and arecalled geodesics. If we were to build a polygon withn-sides, all of which are geodesics, then Eq. (102) reads

2p2(j

Du j5EEM

KdS . (103)

Since the sum of the external angles Du j of a polygon inflat space is 2p, we see that the Gaussian curvature is ameasure of the excess (or deficit) angle in a polygon. Ifthe Gaussian curvature is positive the sum of the exter-nal angles is smaller than 2p so the sum of the internalangles ( j(p2Du j)5(n22)p1**KdS is larger than wemight expect. For example, imagine the triangle on thesphere connecting the North Pole (90 °N), Pontianak,Indonesia (roughly 0 °N, 109 °208W) and Loolo, TheCongo (roughly 0 °S, 19 °208W) along great circles. Ateach vertex of this triangle the arcs of great circles meetat 90° and so the sum of the interior angles is 3p/2. Sincen53 for a triangle, we discover that **KdS5p/2 forthis triangle that covers 1

8 of the globe. However, we


know that for a sphere of radius R , the Gaussian curva-ture is just 1/R2 and so **KdS5 1

8 (4pR2/R2) 5p/2, aswe expect!

There are many beautiful uses and examples of theinterplay between Gaussian curvature and geodesic cur-vature in physical systems. For instance, Avron and Le-vine (Avron and Levine, 1992) have considered dryfoams on curved, two-dimensional surfaces. In this con-text, ‘‘dry’’ refers to the fact that there is no fluid be-tween adjacent bubbles so that the walls between themcan be treated as lines and the vertices may be treated aspoints. There are two key ingredients to the physics ofdry foams: surface tension and pressure. In two dimen-sions the surface tension amounts to a line tension salong the interfaces between neighboring bubbles. If thesoap film is uniform then the line tension is constant aswell. Whenever the lines meet, mechanical equilibriummust be maintained. The pressure in the bubbles exertsa force on their boundaries. If two cells are separated bya boundary line then the pressure difference DP must bebalanced by the boundary. As with the derivation of thewave equation, the force exerted by the boundary curveG is sT8(s). However, the pressure exerts forces onlywithin the surface and so we are only interested in thecomponent of the force in the plane of the surface. Ac-cording to the above discussion, that force isskg(s)g(s), and so the magnitude of the force isskg(s)5DP . In the case of planar foams, kg(s)5k(s)and this is known as the Young-Laplace law. To modelthe diffusion of gas from one cell to the other, we as-sume a simple dynamics where

dN

dt52C(

jDPjlj , (104)

where N is the number of gas molecules in the cell ofinterest, DPj is the pressure difference between it and itsjth neighbor, l j is the length of the boundary separatingthe cell from its jth neighbor, and C.0 is a diffusionconstant. This dynamics captures the simple idea that ifa bubble is higher pressure than its neighbors so thatDPj.0, then it loses gas, while if it has lower pressure itgains gas. Using the generalization of the Young-Laplacelaw and the Gauss-Bonnet theorem we have

dN

dt52Cs R

]Mkg~s !ds

5CsH EEM

KdS1(j

~p2a j!22pJ , (105)

where a j are the internal angles.For flat surfaces with K50 it is known that a hexago-

nal honeycomb network of boundaries minimizes thelength of the cell walls (Morgan, 2000; Hales, 2001). Ifwe imagine starting with a stable configuration on a flatmembrane and distorting the membrane, we should takeeach internal angle to be 2p/3. Doing so Avron and Le-vine (Avron and Levine, 1992) found


dN

dt5CsH p

3~n26 !1EE

MKdSJ . (106)

If we seek a stationary configuration so that N is timeindependent, we see that for flat membranes K50 andonly bubbles with n56 sides are stationary: hexagons. Ifthe surface has positive curvature, K.0, and there is aninstability. To make (dN/dt)50, we see that n,6. How-ever, even if an area and an n were found to make Ntime independent, we can see that there is an instability:if the area grows then (dN/dt) becomes positive so thebubble grows more. Likewise, if the bubble shrinks thenthe integral over K gets smaller and so gas flows out ofthe bubble and it shrinks some more. Thus on a posi-tively curved surface the only stable situation is one forwhich one bubble overtakes the whole system. On theother hand, when the surface has negative Gaussian cur-vature, it follows from a similar argument that when thearea of the bubble increases (dN/dt),0 and when thearea shrinks (dN/dt).0. In the critical case of a flatmembrane the stationary solution is a hexagon and it isneither stable nor unstable towards growth (Avron andLevine, 1992).

B. Euler characteristic and the genus: defects on surfaces

We can take Eq. (102) one step further by consideringa closed surface. In this case we can integrate the Gauss-ian curvature over the whole manifold. If we triangulatethe entire surface to form a net, then we can use theGauss-Bonnet theorem to establish a relation betweenthe topology of the network and the total Gaussian cur-vature. At each vertex there is a total angle of 2p whichis divided into the internal angles of the triangles meet-ing at that point. Each face contributes p to the totalangle at the vertices in addition to the excess angle fromthat face, **KdS . Adding all the triangles together wehave

2pV5pF1TM

KdS , (107)

where V is the number of vertices and F is the numberof faces. Each face contributes three edges E , but eachedge is shared by two triangles, so 3F52E . We thenhave

V2E1F51

2p TM

KdS . (108)

This is a remarkable result. As we show in the bottomtwo graphs of Fig. 10, if we remove an edge from ournetwork (E→E21), two faces join into one (F→F21), and we lose two vertices (V→V22), so V2E1F does not change! Likewise if we add an edge V2E1F is unchanged. Thus our result does not requirethe use of triangles. This invariant is known as the Eulercharacteristic x5V2E1F . Note that if we have anynetwork that can be deformed into a sphere without cut-


ting the edges or changing the vertices, then x can becalculated using Eq. (108) for a perfect sphere, wherethe surface area is 4pR2 and the Gaussian curvature is1/R2. Thus we find that

V2E1F5x52 (109)

for a network with the topology of a sphere. Suppose wetake two faces on the sphere, deform them to be thesame triangle, and place them together, face-to-face. Inthe process we lose three edges (E→E23), two faces(F→F22), and three vertices (V→V23) so x50.What have we done? We have made a doughnut (or atorus) with one handle and have reduced the Euler char-acteristic by 2. Clearly, any time we add a handle x isreduced by 2. If we define the genus g to be the numberof handles of the surface, then

12p T

M

KdS5V2E1F5x52~12g !. (110)

Though the Gaussian curvature was a geometric prop-erty, when integrated over the entire surface it becomesa topological invariant, independent of the local geom-etry.

The Euler characteristic can be used to understandthe topology of defects on closed surfaces. Suppose thatwe have a unit vector field v(x) living in the local tan-gent plane to a closed surface of genus g . Since v(x) liesin the tangent plane, we have v(x)5cos u(x)e1(x)1sin u(x)e2(x), where we have reintroduced our vectorse1(x) and e2(x) which are everywhere perpendicular tothe unit normal n(x) of the surface. We will try to coverthe surface with a vector field that is single valued. Inother words, if we integrate derivatives of v(x) around aclosed curve, we should get back the same vector. But aswe discussed in Sec. III.B, to ensure this property weshould focus on the covariant derivative Du(x). We willhave a single-valued vector field when the curl of thisderivative vanishes. In this case Eqs. (41) and (91) give

TM

@“3“u~x!#•dS5TM

KdS52px . (111)

FIG. 10. Removing or adding an edge from a graph maintainsthe value of the Euler characteristic V2E1F .


Usually “3“u50, but this result tells us that it is nottrue on an arbitrary genus surface M .

Without presenting a detailed discussion on topologi-cal defects (Chaikin and Lubensky, 1995), we can seethat “3“u(x) does not vanish in the presence of a de-fect configuration and that on a two-dimensional sur-face:

“3“u~x!5(i

mid2~x2xi!, (112)

where mi is the charge of the defect and xi is the corre-sponding position. To see this, we consider

u~x!5m tan21S y

x D . (113)

It is straightforward to calculate “u(x) and then to in-tegrate the gradient around a closed curve g that con-tains the origin. We have

2pm5 Rg“u~x!•dx5EE

M“3“u~x!•dS, (114)

where g is the boundary of M . Since we may arbitrarilyshrink the path g around the origin and the integral re-mains constant, it must be that “3“u5md2(x). Gen-eralizing this to an arbitrary collection of defects leadsus to Eq. (112). Using Eq. (112) in Eq. (111) gives us

(i

mi52~12g !5x . (115)

This result is known as the Poincare-Brouwer theorem.Since a vector field must be single valued on the surface,mi must be an integer so that cos u and sin u are welldefined. This result tells us, for instance, that a vectorfield on the surface of a sphere must have two 11 de-fects or one 12 defect. On a torus, however, no defectsare necessary.

There is another, more geometric way to establish Eq.(115). We first argue that two vector fields on the samesurface must have the same total topological charge.Consider two vector fields u(x) and v(x). We can trian-gulate the surface so that each triangle contains at mostone defect in u(x) and one defect of v(x) and that everydefect is in some triangle. In each triangle we may nowintegrate “uu(x) and “uv(x) around the triangleboundary. The total topological charge xu of u(x) is

xu5 (triangles

(edges

E “uu~x!•dx (116)

and similarly for v(x). Note that to establish the anglesuu(x) and uv(x), we must choose basis vectors ei(x).Since we know that “uu(x) depends on our choice ofthese vectors, the integrals of “uu(x) around each tri-angle depend on the underlying geometry through theMermin-Ho relation (41). More importantly, each vectorfield on M may require a different choice of basis vec-tors ei(x), since a defect in any particular triangle forcesa constraint on ei(x). Thus it is not, in general, possible


to calculate xu and xv using the same set of ei(x). How-ever, the difference between xu and xv is just

xu2xv5 (triangles

(edges

E ¹@uu~x!2uv~x!#•dx50.

(117)

The difference vanishes since the angle between u(x)and v(x) is independent of our choice of basis vectors.Therefore when we sum over all triangles we get eachedge twice but in opposite directions. Since we do notneed the basis vectors to define the integrals, it is clearthat the integrals in opposite directions cancel and weget 0. Thus any two vector fields have the same topologi-cal charge.

We will now count the defects of a vector field that weconstruct on triangulated surface, via the following rules:(i) put a defect with 11 charge at each vertex, (ii) put adefect with 11 charge at each center, and (iii) put adefect with 21 charge on each edge. We can see thatthis will produce a consistent vector field everywhereelse, as shown in Fig. 11. Adding together the chargeswe have

xu5V2E1F5x , (118)

and so the topological charge is the Euler characteristicfor any vector field on the surface.

A striking example of the effect of these defects canbe seen when there is a coupling between the bending ofthe surface and the configuration of the vector field(MacKintosh and Lubensky, 1991; Park et al., 1992). Forinstance, if we have a vector field living on a closed sur-face then it must have two 11 defects. But because thedefects are discrete, when we consider Eq. (111) we findthat the Gaussian curvature K must also be confined tothe core of the defect since we may also integrate Eq.(41) over any submanifold of M . But then all the Gauss-ian curvature sits at the defects. If instead of viewing Eq.(41) as a constraint, we can write it as an energy

F512 EE dS A@“u~x!2V~x!#21FCH , (119)

where FCH is the free energy for the surface (93), and A

FIG. 11. A vector field on a triangular patch. Note that the 11defects at the vertices and at the center force 21 defects onthe edges.


is the spin stiffness. On a flat surface we can alwayschoose V50 and so the first term in Eq. (119) just de-scribes the usual Goldstone mode of broken rotationalinvariance. On a curved surface the inclusion of V isnecessary in order to make the free energy uniquely de-fined. The energy (119) now represents a competitionbetween surface bending and the requirement of two11 topological defects which serve as a source forGaussian curvature. Both of these energies vanish forthe plane, but if we restrict our topology to a sphere, forinstance, then neither term can be made to vanish every-where. Balancing these two effects, a vesicle would forman oblate shape, as shown in Fig. 12.

VI. THREE DIMENSIONS AND BEYOND

We have covered the basic elements of the geometryof curves and surfaces. Things become significantly moreabstract in three dimensions because there is no analogof the normal vector, binormal vector, or surface normal:the three-dimensional system is not embedded in ahigher-dimensional space. Though it might be hard tovisualize how something can have curvature without anembedding space, we already know that it is possible.Recall that the Gaussian curvature could be measuredwithout any knowledge of the surface normal. Quanti-ties with that property are called intrinsic. The generaltheory of relativity relies on intrinsic quantities to de-scribe the curvature of four-dimensional space-time(Misner et al., 1973)—there is no reference to a largerspace in which our universe lives. Though we will not gointo any mathematical detail, it is worthwhile to describetwo examples from soft materials.

The first is the blue phases of chiral liquid crystals(Meiboom et al., 1981; Wright and Mermin, 1989). Theseare phases in which there is a three-dimensional, peri-odic modulation of the nematic director n(x) with alength scale comparable to that of visible light. Theseusually exist only over a narrow temperature range andare stabilized by an often neglected term in the nematicfree energy. The Frank free energy for a nematic liquidcrystal is

FIG. 12. A vesicle with some sort of vector order parameter.The Poincare-Brouwer theorem assures that there are two de-fects in the vector field, drawn as the dark lines on the vesicle.The vesicle distorts so that the Gaussian curvature is larger inthe vicinity of the defects.


F512 E d3x$K1~“•n!21K2@n•~“3n!#2

1K3@n3~“3n!#2

12K24“•@~n•“ !n2~“•n!n#%. (120)

The standard elastic constants K1 , K2 , and K3 are ameasure of the energy cost for splay, twist, and bendmodes, respectively (Chaikin and Lubensky, 1995). Thelast term with elastic constant K24 is known as thesaddle-splay term. Though it is a total derivative, whenthere are defects present it can contribute to the energy.The blue phase is riddled with precisely those defectsthat contribute to saddle splay. The remarkable thingabout this term is that it is precisely the Gaussian curva-ture as in Eq. (91):

n~x!•“3V~x!5212“•@~n•“ !n2~“•n!n# . (121)

What does this mean? If there are surfaces to which thenematic director is normal, then the saddle splay is theGaussian curvature of those surfaces. However, thesaddle splay is more general. Even if the director is not afield of layer normals, the saddle splay is a measure ofcurvature. Indeed, the blue phases can be understood asunfrustrated systems in curved three-dimensional space(Sethna et al., 1983). Projecting this texture into flatspace leads to the topological defects in these phases. Acompelling and alternative way of viewing the bluephases is to view them as decorations of space fillingminimal surfaces (Pansu and Dubois-Violette, 1989). Itis the connection between topological defects and curva-ture that makes it possible to identify and locate defectsin lattice simulations of liquid crystals (Priezjev and Pel-covits, 2001).

Another three-dimensional system that can be under-stood in terms of curvature is the melting and freezing ofhard-sphere fluids. In two dimensions, as hard spherescondense from the fluid phase to the crystalline phase,they form close-packed triangles. Eventually these canassemble into a triangular lattice. The situation is not sohappy in three dimensions. There, four spheres can closepack into a tetrahedron, but tetrahedra cannot assembleto fill space. As a result, all lattices have a packing den-sity lower than the best local packing. This difficulty canbe viewed as a geometric frustration along the samelines as the blue phase. In positively curved space, how-ever, this frustration can also be eliminated (Nelson,1983; Nelson and Widom, 1983), and tetrahedral closepacking can fill space.

OUTLOOK

Though the mathematics described here has a certainelegance and beauty, I hope that I have conveyed theutility of differential geometry in a variety of physicalproblems. While a geometric description of a system of-ten leads to an intuitive perspective, there are manyother arenas in which a geometric formulation of the


problem is not only useful but essential. Put togetherwith statistical mechanics, differential geometry hasbeen and will continue to be a powerful tool in the studyof soft materials.

ACKNOWLEDGMENTS

I thank I. Bluestein, B. A. DiDonna, G. M. Grason,M. I. Kamien, W. Y. Kung, T. C. Lubensky, T. R. Powers,P. Ziherl, and especially G. Jungman and P. Nelson forcareful and critical reading of this review. This work wassupported by NSF Grants DMR01-29804 and INT99-10017, and a gift from L. J. Bernstein.

REFERENCES

Aldinger, J., I. Klapper, and M. Tabor, 1995, J. Knot TheoryRamif. 4, 343.

Avron, J. E., and D. Levine, 1992, Phys. Rev. Lett. 69, 208.Bensimon, D., D. Dohmi, and M. Mezard, 1998, Europhys.

Lett. 42, 97.Canham, P., 1970, J. Theor. Biol. 26, 61.Chaikin, P. M., and T. C. Lubensky, 1995, Principles of Con-

densed Matter Physics (Cambridge University Press, Cam-bridge, England).

David, F., 1989, in Statistical Mechanics of Membranes and Sur-faces: Jerusalem Winter School for Theoretical Physics, editedby D. R. Nelson, T. Piran, and S. Weinberg (World Scientific,Singapore).

de Gennes, P.-G., 1970, Scaling Concepts in Polymer Physics(Cornell University Press, Ithaca, NY).

DiDonna, B. A., T. A. Witten, S. C. Venkataramani, and E. M.Kramer, 2002, Phys. Rev. E 65, 016603.

Doi, M., and S. F. Edwards, 1986, The Theory of Polymer Dy-namics (Oxford University Press, New York).

Fain, B., and J. Rudnick, 1999, Phys. Rev. E 60, 7239.Fain, B., J. Rudnick, and S. Ostlund, 1996, Phys. Rev. E 55,

7364.Fuller, F. B., 1971, Proc. Natl. Acad. Sci. U.S.A. 68, 815.Fuller, F. B., 1978, Proc. Natl. Acad. Sci. U.S.A. 75, 3557.Gido, S. P., J. Gunther, E. L. Thomas, and D. Hoffman, 1993,

Macromolecules 26, 4506.Goldstein, H., 1980, Classical Mechanics, 2nd ed. (Addison-

Wesley, Reading, MA).Goldstein, R. E., T. R. Powers, and C. H. Wiggins, 1998, Phys.

Rev. Lett. 80, 5232.Hales, T. C., 2001, Discrete Comput. Geom. 25, 1.Harris, A. B., R. D. Kamien, and T. C. Lubensky, 1999, Rev.

Mod. Phys. 71, 1745.Helfrich, W., 1973, Z. Naturforsch. C 28C, 693.Julicher, F., 1994, Phys. Rev. E 49, 2429.Kamien, R. D., 1998, Eur. Phys. J. B 1, 1.Kamien, R. D., 2001, Appl. Math. Lett. 14, 797.Kamien, R. D., and T. C. Lubensky, 1999, Phys. Rev. Lett. 82,

2892.Kamien, R. D., T. C. Lubensky, P. Nelson, and C. S. O’Hern,

1997, Europhys. Lett. 38, 237.Kleinert, H., 1990, Path Integrals in Quantum Mechanics Sta-

tistics and Polymer Physics (World Scientific, Singapore),Chap. 16.

MacKintosh, F. C., and T. C. Lubensky, 1991, Phys. Rev. Lett.67, 1169.


Marko, J. F., 1997, Europhys. Lett. 38, 183.Marko, J. F., and E. D. Siggia, 1994, Macromolecules 24, 981.Marko, J. F., and E. D. Siggia, 1995, Phys. Rev. E 52, 2912.Meiboom, S., J. P. Sethna, P. W. Anderson, and W. F. Brink-

man, 1981, Phys. Rev. Lett. 46, 1216.Mermin, N. D., and T. L. Ho, 1976, Phys. Rev. Lett. 36, 594.Millman, R. S., and G. D. Parker, 1977, Elements of Differen-

tial Geometry (Prentice-Hall, Englewood Cliffs, NJ).Misner, C. W., K. S. Thorne, and J. A. Wheeler, 1973, Gravi-

tation (Freeman, San Francisco).Morgan, F., 2000, Geometric Measure Theory, 3rd ed. (Aca-

demic Press, San Diego).Moroz, J. D., and R. D. Kamien, 1997, Nucl. Phys. B 506, 695.Moroz, J. D., and P. Nelson, 1997, Proc. Natl. Acad. Sci. U.S.A.

94, 14418.Moroz, J. D., and P. Nelson, 1998, Macromolecules 31, 6333.Nelson, D. R., 1983, Phys. Rev. B 28, 5515.Nelson, D. R., and M. Widom, 1983, Nucl. Phys. B 240, 113.Nelson, P., 1998, Phys. Rev. Lett. 80, 5810.Nitsche, J. C. C., 1989, Lectures on Minimal Surfaces (trans-


lated by J. M. Feinberg) (Cambridge University Press, Cam-bridge, England).

Pansu, B., and E. Dubois-Violette, 1989, Europhys. Lett. 10,43.

Park, J., T. C. Lubensky, and F. C. MacKintosh, 1992, Euro-phys. Lett. 20, 279.

Priezjev, N. V., and R. A. Pelcovits, 2001, Phys. Rev. E 64,031710.

Scherk, H. F., 1835, J. Reine Angew. Math. 13, 185 (also seeNitsche, 1989).

Sethna, J. P., D. C. Wright, and N. D. Mermin, 1983, Phys. Rev.Lett. 51, 467.

Vologodskii, A. V., S. D. Levene, K. V. Klenin, M. Frank-Kamenetskii, and N. R. Cozzarelli, 1992, J. Mol. Biol. 227,1224.

Watson, J. D., and F. H. C. Crick, 1953, Nature (London) 171,737.

White, J. H., 1969, Am. J. Math. 91, 693.Wolgemuth, C. W., T. R. Powers, and R. E. Goldstein, 2000,

Phys. Rev. Lett. 84, 1623.Wright, D. C., and N. D. Mermin, 1989, Rev. Mod. Phys. 61,

385.

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