D IF F E R E N T IA L G EO M ETRY - C O N N E C T IO N S

G i l l i a n T h o r n l e y

(Received January 1991)

Differential geometry, which began with the study of curves and sur­faces, now interacts with topology, algebra, analysis and differential equations in providing a framework for multidimensional mathematics. Its techniques find application in such diverse areas as physics, genetics, control theory, animal science and statistics.

It was the realisation that a surface has its own intrinsic two-dimensional geometry which paved the way for generalisations to multidimensional non-Euclidean geometries. The abstract structures of modern differen­tial geometry still carry some of the names from the classical discipline- tangent space, curvature, torsion - but the concepts have broadened and changed as understanding of the topological foundations grew and it is often difficult to see any connections with the origins of the subject. Manifolds and fibre bundles seem a long way from the classical differ­ential geometry of curves and surfaces. We pick out a few key ideas of Frenet, Gauss, Riemann, Darboux and Cartan as we follow a path from the classical geometry of curves and surfaces to Riemannian geometry in terms of differential forms, and point to some links with fibre bundles.

1. C lassical D ifferential G eom etryClassical differential geometry is concerned with using calculus in the

study of curves and surfaces. The problem of determining tangent lines predates the calculus and was a motivating factor in the development of the derivative. Thus the origins of the subject are inextricably inter­twined with those of calculus itself.

The shape of a curve in three dimensional space is most easily de­scribed in terms of the changes in the tangent and normal vectors as we move along the curve. At each point on the curve we have a unit tangent vector, T, a principal normal vector, N (the unit vector in the direction T ') and the binormal vector, B, which is orthogonal to both T and N.

These give rise to three vector fields along the curve, providing an orthonormal basis at each point of the curve. The curvature, k , and torsion, r , are associated with the rate of change of the tangent and

Math. Chronicle 20 (1991), 2 7 -3 7

binormal vector fields and (when the arc length is the parameter) are given by the Frenet formulae (1847):

T - kN N ' = - kT + t B B ' = —t N.

The concept of curvature was well known before the time of Frenet whose significant contribution is the choice of the vector fields T, N , B as frames of reference. This idea is the forerunner of C artan’s method of “moving frames” and the concept of the frame bundle.

The shape of a surface in three-dimensional space is less easy to quan­tify. In the eighteenth century Monge, Euler and Meusnier approached it by way of the curve of intersection of a plane and the surface. For simplicity we choose a plane containing the surface normal, ra, at a point P and a vector, v, which is tangent to the surface at P (i.e. v is tangent to some curve through P which lies in the surface). This plane cuts the surface in a curve called the normal section of the surface at P in the direction of v.

The curvature of the normal section is often called the normal curva­ture k(v), of the surface in the direction v. This produces a measure of curvature at P for each direction in the plane tangent to the surface at P.

Euler (1760) showed that when the normal curvatures at P are not constant there is exactly one direction in which k(v) has a maximum value and one direction in which it has a minimum value, and these directions (called principal directions) are orthogonal. The maximum and minimum values of k(v) are called principal curvatures.

It was another fifty years before Gauss, in his paper of 1827, provided a fresh approach which avoided dissecting the surface into curves.

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He realised that the shape of the surface near the point P could be de­scribed by the change in the directions of the surface normals at neigh­bouring points. He measured this change using the ‘Gauss m ap’, G, which maps a region of the surface M to the unit sphere The point P in M is mapped to the point G(P) whose position vector is equal to the unit surface normal, n, at P.

A neighbourhood U of a point P on a smooth surface is mapped to a region G(U) on For his curvature Gauss considered the ratio of the area of G(R) to the area of R where R is a “surface element” having infinitely small area. His curvature is therefore some sort of limit, as the neighbourhood U collapses to P, of

area of G(U) area of U

It turns out that this Gaussian curvature is actually the product of the maximum and minimum values of the normal curvatures at P (see [3])-

The truly remarkable result which Gauss proved (his Theorema Egregium) was that his measure of curvature is independent of the imbedding in three-dimensional Euclidean space and so is intrinsic to the surface. A curved surface therefore has its own intrinsic two-dimensional geome­try. This fact led Riemann to the generalisations to higher dimensional non-Euclidean geometries in his inaugural lecture of 1854 at Gottingen.

2. R iem annRiemann was subsequently to occupy the chair of mathematics at

Gottingen but in 1854 Gauss was still the professor and Riemann was seeking the lowliest academic position. It gave the right to lecture and to collect fees from the students attending the lectures but didn’t actually provide a salary. To qualify for this position the candidate had to write a paper and give a lecture to the faculty, the topic for the lecture being chosen by the faculty from a list of three submitted by the candidate.

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Riemann’s doctoral thesis had been on the foundations for a general theory of functions of a complex variable, and at the time he was assist­ing Weber in the physics laboratory investigating connections between electricity, magnetism, light and gravitation. So his first two topics related to these areas and for a third he put down The Foundations of Geometry. Gauss had been impressed with Riemann’s creative and original work in complex variable theory but geometry had always held a particular interest for him and he was curious to see what Riemann would do with it. So it was that the faculty chose Riemann’s third topic, geometry. It took Riemann some months to prepare the lecture (he was ill part of the time) and he went to a great deal of trouble to make it intelligible for the faculty of philosophy audience which included many who were unfamiliar with mathematics.

The lecture therefore discussed some of the most influential ideas in the history of differential geometry in general philosophical terms and suppressed the analytic investigations and formulae. In it Riemann pointed out the need to separate topological properties from metric ones. He was able to clarify the confusion over the status of non-Euclidean geometry and introduced a wide class of non-Euclidean spaces with dif­ferent curvatures. He attem pted to define a manifold which he char­acterised as being locally like Euclidean space and it is clear that he was aware of the importance of infinite-dimensional spaces such as the set of all real-valued functions on a space. He discussed the problem of assigning lengths to curves using an integral and took as an example the simplest case where s = f f and / = y/gijdxidxj is positive defi­nite. He also considered the conditions for two Riemannian manifolds to be locally isometric and through some counting arguments arrived at results which were not proved rigorously for another hundred years. ([3] and [4]).

Gauss was impressed with the depth of the ideas and Riemann got his job, but the lecture was not published until after his death in 1866. Although Helmholtz and Clifford responded to Riemann’s geometry the work was not generally appreciated until the development of tensor cal­culus in the early twentieth century and Einstein’s work in relativity.

3. M anifoldsA surface is a 2-manifold and is locally like a piece of the Euclidean

plane. An n-dimensional manifold M is a topological Hausdoff space which is locally like a piece of lRn in the sense that each point P in M has a neighbourhood which is homeomorphic to an open set in IR™.

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I1

The homeomorphisms provide “co-ordinate functions” on the neigh­bourhoods on the manifold. Where two neighbourhoods overlap the homeomorphisms provide mappings of IRn to itself (for example, 0 o ^>- 1

and ip o and these must be differentiable functions. This gives the manifold a differentiable structure and other geometric structures such as connections, metric, complex structure, foliations etc., may be added. The ‘local’ approach to differential geometry tends to use co-ordinates on the neighbourhoods and is interested in the properties of the mani­fold which are invariant under co-ordinate transformations. The ‘global’ approach studies the manifold as a whole and seeks global invariants.

A curve in a manifold M is simply a mapping from an interval I of the real numbers to the manifold.

c : I — M t —+ c(<)

The manifold automatically has tangent vectors associated with it, namely the vectors c'(t) which are tangent to curves in the manifold. The set of all tangent vectors at a point P is called the iangeni space at P. If we include the zero vector, this is a vector space and there is a dual vector space associated with it. Elements of the dual space are linear functions which map the vectors back to the field of scalars. These are the 1-forms or covariant vectors.

From these two vector spaces we can form tensor products and the elements of the tensor product spaces are tensors. Ricci and Levi Civita (1901) developed a tensor calculus which remains a prominent feature of many applications of differential geometry in physics and in statisitics.

Darboux (1880) adapted Frenet’s approach to curves to the study of surfaces and this method was further developed and extended by Elie C artan this century in his method of moving frames and differen­tial forms. The idea is to assign an ordered orthonormal basis e\(P),

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e2 ( P ) . . . en(P ) to the tangent space at each point P of a region in the n-manifold. This gives n vector fields ej, e%... en over the region. Together they constitute a field of frames. For example, in Euclidean 3- space the field of natural frames assigns to each point the three unit vectors i, j , k in the directions of the co-ordinate axes.

A structure called a connection is introduced (historically its role was described as connecting the tangent spaces at neighbouring points). It appears in the definition of a covariant derivative. This derivative gives the change in the basis vector field e* as we move in a direction v to a neighbouring point. It is written in terms of the basis vector field as

n

V v ei = Y ^ V i j ('>)ej ( 1 )j =i

where u>ij are 1-forms, the connection forms of the frame.To develop the theory in terms of differential forms we turn to the

dual vector spaces of 1-forms. Corresponding to the basis e\, . . . en of the tangent space, the dual vector space has a basis 6\, do ■ where

ei(ei) = 6ij = { h iU = {1 0 , otherwise

and the 0,- are called the dual 1 -forms of the frame e,. In Euclidean space the dual 1-forms satisfy 0i(v) — v • e, where v is a tangent vector field. The dual 1-forms of the natural frame i, j, k are the differentials dx, dy, dz.

In order to establish the structural equations of Euclidean space we need two operations on differential forms1. The exterior product, A, of two 1-forms which produces a 2-form and

satisfiesa A /3(u, v) = a(u)/3(v) — a(v)(3(u).

We note that a A )3 = —(3 A a.

2. The exterior derivative, d, which maps 1-forms into 2-forms and sat­isfies

d(a A (3) = da A (3 — aA d/3.

It can be shown that second derivatives are always zero so d(da) = 0.

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The connection equations (1) have a dual version in terms of the dual 1 -forms, namely,

n

dOi = ^ ujij A 6j , i = 1 . . . n. (2)i = 1

By taking the exterior derivative of these equations and using (2) to­gether with the independence of the dual 1-forms we obtain a second set of equations:

ndujij = (jjik A Ukj, i, j = 1 • . - n. (3)

jfc=i

Equations (2) and (3) are the structural equations of Euclidean space. For orthonormal frames the connection forms u>,j are skew symmetric and are completely determined by (2). We will show that equations (3) are associated with curvature by considering the special case of a surface in 3-dimensional Euclidean space, E 3.

4. Surfaces in three-d im ensional Euclidean Space

On a smooth oriented surface in E 3 we choose a frame field e\, e2, e3 having e3 equal to the unit vector field n normal to the surface.

e \ , e2 is then a frame field for vectors tangent to the surface. Suppose v is a vector field tangent to the surface. Since it is also a vector field inE 3,

0 3 (u) = v • c3 = v • n = 0 .

If we now restrict our attention to the surface and its tangent vectors we can take as being zero and the structural equations (2) and (3)

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giye

ddi = ^ 2 <*>ij A Qj, * = 1 ,2 (4)j =i

d63 = 0 = W31 A + W32 A d2 (5)

< ^ 1 2 — wi3 A W32 (6 )

(Note that u>a = 0 from the skew symmetry of the connection forms.) Equations (4) are structural equations for the 2-dimensional geometry of the surface. We will show that the other equations can be linked to the Gaussian curvature.

O’Neill, in [2], defines a shape operator on the surface by

S(v) = - V „ n

where v is a tangent vector to the surface and n the unit surface normal. It follows that the normal curvature, k(v) = S(v) • v. Through Euler’s results on principal curvatures (see §1 ) it is possible to show that the Gaussian curvature K = d e tS , ([2]).

Now we have n — C3 ,

3

so S(v) = — VPc3 = — uj3j (v)ej from ( 1 )j = 1

= -W3l(«0«l -U>32(*>)«?2-

The matrix representation of S relative to the basis e \ , e2 is

^ 1 3(^1 ) ^ 1 3 (^2 )^ 2 3 (^1 ) w23(e2)

HenceK = det S = w 1 3 (0 1 )0 2̂ 3 (^2 ) ~ ^i3(^2)w23(^i)

= W13 Au;23(ei,e2)

= -du> i 2 (e i,« 2) fr°m (6)-

It follows that du}\2 = —K0i A d2 and the final structural equation (6 ) contains the curvature.

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5. Structural Equations for an n-M anifoldGiven a frame field with dual 1 -forms 0,-, i = 1 . . . n , the connection

forms, ujij, are defined by the first structural equations:

n

d6{ — ^ ̂u>ij A 9j , i — 1 . . . n, u>ij = u)ji. j = i

A curvature 2-form is defined by the second structural equations:

n

du>ij ='^2u>ik A u k j i , j = l . . . n . (7)*=i

From equation (3) we see that the curvature 2-form is zero in Euclidean space. For n = 2, equation (7) reduces to dw 12 = ^ i 2 > and for the surface in E 3,

£7i2 = —K0\ A 62

where K is the Gaussian curvature.The familiar Riemannian geometry follows from these structural equa­

tions using the natural frame associated with a co-ordinate system. di — d fd x i replaces e* and dxi replaces 9i. The metric tensor is the inner product of the basis vectors (di,dj). The connection forms u>ij, being 1-forms, can be written in terms of the basis for 1 -forms, {dx{, i = 1 . . . n}

nu>ij = T Fjjkdxk

k — \

and Ttjik are Christoffel symbols. The curvature 2-form

&ij = ^R ijhkdxh A dxk

where Rijhk is the Riemann curvature tensor.

6. B undles over a M anifoldA manifold M has a vector space of tangent vectors associated with

each point. The collection of all tangent vectors at all points of M can be thought of as a new manifold T M called the tangent bundle over M . This is illustrated diagramatically as follows:

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The projection, 7r : T M M , maps a tangent vector v(P) to the point P in the manifold M . The fibre over P, ir~l (P), is the tangent space at P. A section of the bundle is a mapping s : M —* T M such that 7ro s is the identity on M. A vector field on M is therefore a section of the tangent bundle since it assigns a tangent vector to each point of M.

The bundle of frames, F M , over M consists of the collection of all ordered bases for all tangent spaces of M . An element of F M is a basis for the tangent space at a particular point of M.

The projection n : F M —► M maps a frame for the tangent space at P to the point P. The fibre over P, n~1(P), consists of all bases for the tangent space at P. Two elements in F M belong to the same fibre if and only if they are bases for the same vector space. This means that one can be obtained from the other by a non-singular linear transformation. The fibre is therefore isomorphic to the general linear group GL(n, IR). This bundle is a differentiable principal G-bundle over M with group GL(n,JR).

A section of the bundle is a mapping, s : M —► F M such that 7r o s is the identity on M . It assigns a frame to each point of M and so C artan’s “moving frame” is a section of this bundle of frames.

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R e f e r e n c e s

1. C. Lanczos, Space through the Ages, Academic Press, 1970.2. B. O ’Neill, Elementary Differential Geometry, Academic Press, 1966.3. M. Spivak, A Comprehensive Introduction to Differential Geometry,

Volume II, Publish or Perish, 1979.4. R. Torretti, Philosophy of Geometry from Riemann to Poincare,

Reidel, 1978.5. I.M. Yaglom, Felix Klein and Sophus Lie, Birkhauser, 1988.

Gillian Thom ley,Massey University,Palmerston North,NEW ZEALAND.

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