geometry of warped products as riemannian submanifolds and related

SOOCHOW JOURNAL OF MATHEMATICS

Volume 28, No. 2, pp. 125-156, April 2002

GEOMETRY OF WARPED PRODUCTS AS RIEMANNIAN

SUBMANIFOLDS AND RELATED PROBLEMS

BY

BANG-YEN CHEN

Abstract. The warped product N1 ×f N2 of two Riemannian manifolds (N1, g1)

and (N2, g2) with warping function f is the product manifold N1×N2 equipped with

the warped product metric g1+f2g2, where f is a positive function on N1. It is well-

known that the notion of warped products plays some important roles in differential

geometry as well as in physics. In this article we survey recent results on warped

products isometrically immersed in real or complex space forms as Riemannian,

Lagrangian, or CR-submanifolds. Moreover, we also present some recent related

results concerning convolution of Riemannian manifolds, constant-ratio submani-

folds, T -constant submanifolds, N -constant submanifolds and rectifying curves.

1. Introduction

Let B and F be two Riemannian manifolds of positive dimensions equipped

with Riemannian metrics gB and gF , respectively, and let f be a positive function

on B. Consider the product manifold B × F with its projection π : B × F → B

and η : B × F → F . The warped product M = B ×f F is the manifold B × F

equipped with the Riemannian structure such that

||X||2 = ||π∗(X)||2 + f2(π(x))||η∗(X)||2, (1.1)

Communicated November 10 2001.AMS Subject Classification. 53C40, 53C42, 53C50.Key words. warped product, warped product immersion, CR-product, Segre embedding,

CR-warped product, constant-ratio submanifold, T -constant submanifold, N -constant subman-ifold, rectifying curve, convolution manifold, convolution Riemannian manifold.

This is the written version of author’s talk delivered at the International Conference onDifferential Geometry held at Tokyo Metropolitan University, December 17-19, 2001, celebratingthe 60th anniversary of Professor Koichi Ogiue.

125

126 BANG-YEN CHEN

for any tangent vector X ∈ TxM . Thus, we have g = gB + f2gF . The function f

is called the warping function of the warped product (cf. [4]). It is well-known

that the notion of warped products plays some important roles in differential

geometry as well as in physics ([2, 49]). For instance, the best relativistic model

of the Schwarzschild space-time that describes the out space around a massive

star or a black hole is given as a warped product (cf. [49], pp.364-367).

One of the most fundamental problems in the theory of submanifolds is the

immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean

space (or, more generally, in a space form). According to a well-known theorem

of J. F. Nash, every Riemannian manifold can be isometrically immersed in some

Euclidean spaces with sufficiently high codimension. Nash’s theorem implies in

particular that every warped product N1×f N2 can be immersed as a Riemannian

submanifold in some Euclidean space. Moreover, many important submanifolds

in real and complex space forms are expressed as warped products submanifolds,

see, for instance, ([1, 4, 5, 6, 12, 14, 36, 37, 38, 39, 49, 50]).

In view of these simple facts, we propose the following very simple problem:

Problem 1.

∀ N1 ×f N2

isometric

−−−−−−−−→immersion

Em or Rm(c) =⇒ ??? (1.2)

where Rm(c) denotes a Riemannian m-manifold of constant sectional curvature

c.

In particular, we ask the following.

Problem 1.1. Given a warped product N1 ×f N2, what are the necessary

conditions for the warped product to admit a minimal isometric immersion in a

Euclidean m-space Em (or in Rm(c))?

Problem 1.2. Let N1 ×f N2 be an arbitrary warped product immersed in

a Euclidean space Em as a Riemannian submanifold. What are the relationships

between the warped product structure of N1 ×f N2 and extrinsic structures of

N1 ×f N2 in Em (or in Rm(c)) as a Riemannian submanifold ?

Notice that the main warped structure of a warped product is warping func-

tion.

GEOMETRY OF WARPED PRODUCTS AS RIEMANNIAN SUBMANIFOLDS 127

In this article, we present recent results and related results concerning these

basic problems. More precisely, we survey recent results concerning warped prod-

ucts isometrically immersed in real or complex space forms as Riemannian, La-

grangian, or CR-submanifolds. Moreover, we also present some related results

on convolution of Riemannian manifolds, constant-ratio submanifolds, T -constant

submanifolds, N -constant submanifolds, as well as rectifying curves.

2. Basic Formulas

Let N be an n-dimensional submanifold of a Riemannian m-manifold Rm(c)

of constant sectional curvature c. We choose a local field of orthonormal frame

e1, . . . , en, en+1, . . . , em in Rm(c) such that, restricted to N , the vectors e1, . . . , en

are tangent to N and en+1, . . . , em are normal to N .

Let K(ei ∧ ej), 1 ≤ i < j ≤ n, denote the sectional curvature of the plane

section spanned by ei, ej . Then the scalar curvature of N is given by

τ =∑

i<j

K(ei ∧ ej). (2.1)

For a submanifold N in Rm(c) we denote by ∇ and ∇ the Levi-Civita con-

nections of N and Rm(c), respectively. The Gauss and Weingarten formulas are

given respectively by

∇XY = ∇XY + σ(X,Y ), (2.2)

∇Xξ = −AξX +DXξ, (2.3)

for vector fieldsX,Y tangent toN and vector field ξ normal toN , where σ denotes

the second fundamental form, D the normal connection, and A the shape operator

of the submanifold. Let {σrij}, i, j = 1, . . . , n; r = n+ 1, . . . ,m, denote the coef-

ficients of the second fundamental form with respect to e1, . . . , en, en+1, . . . , em.

The mean curvature vector−→H is defined by

−→H =

1

ntrace σ =

1

n

n∑

i=1

σ(ei, ei), (2.4)

where {e1, . . . , en} is a local orthonormal frame of the tangent bundle TN of

N . The squared mean curvature is given by H2 = 〈 −→H,−→H 〉, where 〈 , 〉 denotes

128 BANG-YEN CHEN

the inner product. A submanifold N is called minimal in Rm(c) if the mean

curvature vector of N in Rm(c) vanishes identically.

Denote by R the Riemann curvature tensors of N . Then the equation of

Gauss is given by (see, for instance, [7])

R(X,Y ;Z,W ) = c{〈X,W 〉〈Y,Z〉 − 〈X,Z〉〈Y,W 〉}+ 〈σ(X,W ), σ(Y,Z)〉 − 〈σ(X,Z), σ(Y,W )〉 , (2.5)

for vectors X,Y,Z,W tangent to N .

Let M be a Riemannian p-manifold and e1, . . . , ep be an orthonormal frame

fields on M . For differentiable function ϕ on M , the Laplacian ∆ϕ of ϕ is defined

by

∆ϕ =p

∑

j=1

{(∇ejej)ϕ− ejejϕ}. (2.6)

3. Riemannian Products, Moore’s Lemma and Nolker’s Theorem

Suppose that M1, . . . ,Mk are Riemannian manifolds and that

f : M1 × · · · ×Mk → EN

is an isometric immersion of the Riemannian productM1×· · ·×Mk into Euclidean

N -space. J. D. Moore [42] proved that if the second fundamental form h of f

has the property that h(X,Y ) = 0 for X tangent to Mi and Y tangent to Mj,

i 6= j, then f is a product immersion, that is, there exist isometric immersions

fi : Mi → Emi , 1 ≤ i ≤ k, such that

f(x1, . . . , xk) = (f(x1), . . . , f(xk)), (3.1)

when xi ∈Mi for 1 ≤ i ≤ k.

Let M0, · · · ,Mk be Riemannian manifolds, M = M0×· · ·×Mk their product,

and πi : M →Mi the canonical projection. If ρ1, · · · , ρk : M0 → R+ are positive-

valued functions, then

〈X,Y 〉 := 〈π0∗X,π0∗Y 〉 +k

∑

i=1

(ρi ◦ π0)2 〈πi∗X,πi∗Y 〉 (3.2)


defines a Riemannian metric on M , called a warped product metric. M endowed

with this metric is denoted by M0 ×ρ1M1 × · · · ×ρk

Mk.

A warped product immersion is defined as follows: LetM0×ρ1M1×· · ·×ρk

Mk

be a warped product and let fi : Ni →Mi, i = 0, · · · , k, be isometric immersions,

and define σi := ρi ◦ f0 : N0 → R+ for i = 1, · · · , k. Then the map

f : N0 ×σ1N1 × · · · ×σk

Nk →M0 ×ρ1M1 × · · · ×ρk

Mk (3.3)

given by f(x0, · · · , xk) := (f0(x0), f1(x1), · · · , fk(xk)) is an isometric immersion,

which is called a warped product immersion.

S. Nolker [47] extended Moore’s result to the following.

Nolker Theorem. Let f : N0×σ1N1 ×· · ·×σk

Nk → RN (c) be an isometric

immersion into a Riemannian manifold of constant curvature c. If h is the sec-

ond fundamental form of f and h(Xi, Xj) = 0, for all vector fields Xi and Xj,

tangent to Ni and Nj respectively, with i 6= j, then, locally, f is a warped product

immersion.

Warped product immersions was applied in [38] by Dillen and Nolker to clas-

sify semi-parallel immersions with flat normal connection. For a recent extensions

of Nolker theorem, see [42, 51].

4. Warped Products in Real Space Forms

In this section we present some recent general results concerning warped

products isometrically immersed in real space forms. These results provide some

solutions to Problem 1.

Theorem 4.1.([28]) For any isometric immersion φ : N1 ×f N2 → Rm(c) of

a warped product N1 ×f N2 into a Riemannian manifold of constant curvature c,

we have∆f

f≤ n2

4n2H2 + n1c, (4.1)

where ni = dimNi, n = n1 +n2, H2 is the squared mean curvature of φ, and ∆f

is the Laplacian f on N1.

130 BANG-YEN CHEN

The equality sign of (4.1) holds identically if and only if there exists a warped

product representation M1 ×ρ M2 of Rm(c) such that φ : N1 ×f N2 →M1 ×ρ M2

is a minimal warped product immersion.

Theorem 4.2.([27]) For any isometric immersion φ : N1 ×f N2 → Rm(c),

the scalar curvature τ of the warped product N1 ×f N2 satisfies

τ ≤ ∆f

n1f+n2(n− 2)

2(n− 1)H2 +

1

2(n+ 1)(n− 2)c. (4.2)

If n = 2, the equality case of (4.2) holds automatically.

If n ≥ 3, the equality sign of (4.2) holds identically if and only if either

(a) N1 ×f N2 is of constant curvature c, the warping function f is an eigen-

function with eigenvalue c, i.e., ∆f = cf , and N1 ×f N2 is immersed as a totally

geodesic submanifold in Rm(c), or

(b) locally, N1 ×f N2 is immersed as a rotational hypersurface in a totally

geodesic submanifold Rn+1(c) of Rm(c) with a geodesic of Rn+1(c) as its profile

curve.

Remark 4.1. Every Riemannian manifold of constant curvature c can be

locally expressed as a warped product whose warping function satisfies ∆f = cf .

For examples, Sn(1) is locally isometric to (0,∞) ×cos x Sn−1(1); E

n is locally

isometric to (0,∞) ×x Sn−1(1); and Hn(−1) is locally isometric to R ×ex E

n−1.

Remark 4.2. There are other warped product decompositions of Rn(c)

whose warping function satisfies ∆f = cf . For example, let {x1, . . . , xn1} be a

Euclidean coordinate system of En1 and let ρ =

∑n1

j=1 ajxj+b, where a1, . . . , an1, b

are real numbers satisfying∑n1

j=1 a2j = 1. Then the warped product E

n1 ×ρSn2(1)

is a flat space whose warping function is a harmonic function.

In fact, those are the only warped product decompositions of flat spaces whose

warping functions are harmonic functions.

Remark 4.3. Ejiri constructed in [39] many examples of minimal warped

product immersions into Riemannian manifolds of constant curvature. Moreover,

he also proved the following.


Theorem 4.3.([40]) There exist countably many immersions of S1 × Sn−1

into Sn+1 such that the induced metric on S1 × Sn−1 is a warped product of

constant scalar curvature n(n− 1).

Now, we provide some immediate applications.

Corollary 4.1.([28]) Let N1 and N2 be two Riemannian manifold and f be

a nonzero harmonic function on N1. Then every minimal isometric immersion

of N1 ×f N2 into any Euclidean space is a warped product immersion.

Remark 4.4. There exist many such minimal immersions. For example, if

N2 is a minimal submanifold of Sm−1(1) ⊂ Em, then the minimal cone C(N2)

over N2 with vertex at the origin is a warped product R+ ×s N2 whose warping

function f = s is a harmonic function. Here s is the coordinate function of the

positive real line R+.

Corollary 4.2.([28]) Let f 6= 0 be a harmonic function on N1. Then for

any Riemannian manifold N2 the warped product N1 ×f N2 does not admits any

minimal isometric immersion into any hyperbolic space.

Corollary 4.3.([28]) Let f be a function on N1 with ∆f = λf with λ > 0.

Then for any Riemannian manifold N2 the warped product N1 ×f N2 admits no

minimal isometric immersion into any Euclidean space or hyperbolic space.

Remark 4.5. In views of Corollaries 4.2 and 4.3, we point out that there

exist minimal immersions from N1 ×f N2 into hyperbolic space such that the

warping function f is an eigenfunction with negative eigenvalue. For example,

R ×ex En−1 admits an isometric minimal immersion into the hyperbolic space

Hn+1(−1) of constant curvature −1.

Corollary 4.4.([28]) If N1 is a compact, then N1 ×f N2 does not admit a

minimal isometric immersion into any Euclidean space or hyperbolic space.

Remark 4.6. For Corollary 4.4, we point out that there exist many minimal

immersions of N1 ×f N2 into Em with compact N2. For examples, a hypercater-

noid in En+1 is a minimal hypersurfaces which is isometric to a warped product

R ×f Sn−1. Also, for any compact minimal submanifold N2 of Sm−1 ⊂ E

m, the

minimal cone C(N2) is a warped product R+ ×sN2 which is also a such example.

132 BANG-YEN CHEN

Remark 4.7. Contrast to Euclidean and hyperbolic spaces, Sm admits

minimal warped product submanifolds N1 ×f N2 with N1 and N2 being compact.

The simplest such examples are minimal Clifford tori defined by

Mk,n−k = Sk

√

k

n

× Sn−k

√

n− k

n

⊂ Sn+1, 1 ≤ k < n.

Remark 4.8. In [54] Suceava construct a family of warped products of

hyperbolic planes which do not admit any isometric minimal immersion into

Euclidean space, by applying the invariants introduced in [15].

5. Warped Product Real Hypersurfaces in Complex Space Forms

For Riemannian product real hypersurfaces in complex space forms, we have

the following non-existence theorem.

Theorem 5.1.([32]) There do not exist real hypersurfaces in complex pro-

jective and complex hyperbolic spaces which are Riemannian products of two or

more Riemannian manifolds of positive dimension.

In other words, every real hypersurface in a nonflat complex space form is

irreducible.

A contact manifold is an odd-dimensional manifold M 2n+1 with a 1-form η

such that η ∧ (dη)n 6= 0. A curve γ = γ(t) in a contact manifold is called a

Legendre curve if η(β ′(t)) = 0 along β. Let S2n+1(c) denote the hypersphere

in Cn+1 with curvature c centered at the origin. Then S2n+1(c) is a contact

manifold endowed with a canonical contact structure which is the dual 1-form

of the characteristic vector field Jξ, where J is the complex structure and ξ the

unit normal vector on S2n+1(c).

Legendre curves are known to play an important role in the study of contact

manifolds, e.g. a diffeomorphism of a contact manifold is a contact transformation

if and only if it maps Legendre curves to Legendre curves.

Contrast to Theorem 5.1, there exist many warped product real hypersurfaces

in complex space forms as given in the following three theorems.


Theorem 5.2.([24]) Let a be a positive number and γ(t) = (Γ1(t),Γ2(t)) be

a unit speed Legendre curve γ : I → S3(a2) ⊂ C2 defined on an open interval I.

Then

x(z1, . . . , zn, t) = (aΓ1(t)z1, aΓ2(t)z1, z2, . . . , zn), z1 6= 0, (5.1)

defines a real hypersurface which is the warped product Cn∗∗ ×a|z1| I of a complex

n-plane and I, where Cn∗∗ = {(z1, . . . , zn) : z1 6= 0}.

Conversely, up to rigid motions of Cn+1, every real hypersurface in C

n+1

which is the warped product N ×f I of a complex hypersurface N and an open

interval I is either obtained in the way described above or given by the product

submanifold Cn × C ⊂ C

n × C1 of C

n and a real curve C in C1.

Let S2n+3 denote the unit hypersphere in Cn+2 centered at the origin and

put U(1) = {λ ∈ C : λλ = 1}. Then there is a U(1)-action on S2n+3 defined by

z 7→ λz. At z ∈ S2n+3 the vector V = iz is tangent to the flow of the action.

The quotient space S2n+3/ ∼, under the identification induced from the action,

is a complex projective space CP n+1 which endows with the canonical Fubini-

Study metric of constant holomorphic sectional curvature 4. The almost complex

structure J on CP n+1 is induced from the complex structure J on Cn+2 via the

Hopf fibration: π : S2n+3 → CP n+1. It is well-known that the Hopf fibration π

is a Riemannian submersion such that V = iz spans the vertical subspaces.

Let φ : M → CP n+1 be an isometric immersion. Then M = π−1(M) is a

principal circle bundle over M with totally geodesic fibers. The lift φ : M →S2n+3 of φ is an isometric immersion so that the diagram:

M CP n+1

M S2n+3

................................................................................................................

...............

φ

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π

commutes.

Conversely, if ψ : M → S2n+3 is an isometric immersion which is invariant

under the U(1)-action, then there is a unique isometric immersion ψπ : π(M ) →

134 BANG-YEN CHEN

CP n+1 such that the associated diagram commutes. We simply call the immer-

sion ψπ : π(M ) → CP n+1 the projection of ψ : M → S2n+3.

For a given vector X ∈ Tz(CPn+1) and a point u ∈ S2n+2 with π(u) = z,

we denote by X∗u the horizontal lift of X at u via π. There exists a canonical

orthogonal decomposition:

TuS2n+3 = (Tπ(u)CP

n+1)∗u ⊕ Span {Vu}. (5.2)

Since π is a Riemannian submersion, X and X∗u have the same length.

We put

S2n+1∗ =

{

(z0, . . . , zn) :n

∑

k=0

zkzk = 1, z0 6= 0}

, CP n0 = π(S2n+1

∗ ). (5.3)

Theorem 5.3.([24]) Suppose that a is a positive number and γ(t) = (Γ1(t),

Γ2(t)) is a unit speed Legendre curve γ : I → S3(a2) ⊂ C2 defined on an open

interval I. Let x : S2n+1∗ × I → C

n+2 be the map defined by

x(z0, . . . , zn, t) = (aΓ1(t)z0, aΓ2(t)z0, z1, . . . , zn),n

∑

k=0

zkzk = 1. (5.4)

Then

(i) x induces an isometric immersion ψ : S2n+1∗ ×a|z0| I → S2n+3.

(ii) The image ψ(S2n+1∗ ×a|z0| I) in S2n+3 is invariant under the action of U(1).

(iii) the projection ψπ : π(S2n+1∗ ×a|z0|I) → CP n+1 of ψ via π is a warped product

hypersurface CP n0 ×a|z0| I in CP n+1.

Conversely, if a real hypersurface in CP n+1 is a warped product N ×f I of

a complex hypersurface N of CP n+1 and an open interval I, then, up to rigid

motions, it is locally obtained in the way described above.

In the complex pseudo-Euclidean space Cn+21 endowed with pseudo-Euclidean

metric

g0 = −dz0dz0 +n+1∑

j=1

dzjdzj , (5.5)

we define the anti-de Sitter space-time by

H2n+31 = {(z0, z1, . . . , zn+1) : 〈z, z〉 = −1}. (5.6)


It is known that H2n+31 has constant sectional curvature −1. There is a U(1)-

action on H2n+31 defined by z 7→ λz. At a point z ∈ H2n+3

1 , iz is tangent to the

flow of the action. The orbit is given by zt = eitz with dzt

dt= izt which lies in

the negative-definite plane spanned by z and iz. The quotient space H 2n+31 / ∼

is the complex hyperbolic space CHn+1 which endows a canonical Kahler metric

of constant holomorphic sectional curvature −4. The complex structure J on

CHn+1 is induced from the canonical complex structure J on Cn+21 via the totally

geodesic fibration: π : H2n+31 → CHn+1.

Let φ : M → CHn+1 be an isometric immersion. Then M = π−1(M) is a

principal circle bundle over M with totally geodesic fibers. The lift φ : M →H2n+3

1 of φ is an isometric immersion such that the diagram:

M CHn+1

M H2n+31

................................................................................................................

...............

φ

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π

commutes.

Conversely, if ψ : M → H2n+31 is an isometric immersion which is invariant

under the U(1)-action, there is a unique isometric immersion ψπ : π(M ) →CHn+1, called the projection of ψ so that the associated diagram commutes. We

put

H2n+11∗ = {(z0, . . . , zn) ∈ H2n+1

1 : zn 6= 0}, (5.7)

CHn∗ = π(H2n+1

1∗ ). (5.8)

Theorem 5.4.([24]) Suppose that a is a positive number and γ(t) = (Γ1(t),

Γ2(t)) is a unit speed Legendre curve γ : I → S3(a2) ⊂ C2. Let y : H2n+1

1∗ × I →C

n+21 be the map defined by

y(z0, . . . , zn, t) = (z0, . . . , zn−1, aΓ1(t)zn, aΓ2(t)zn), (5.9)

z0z0 −n

∑

k=1

zkzk = 1. (5.10)

Then

136 BANG-YEN CHEN

(i) y induces an isometric immersion ψ : H2n+11∗ ×a|zn| I → H2n+3

1 .

(ii) The image ψ(H2n+11∗ ×a|zn| I) in H2n+3

1 is invariant under the U(1)-action.

(iii) the projection ψπ : π(H2n+11∗ ×a|zn| I) → CHn+1 of ψ via π is a warped

product hypersurface CHn∗ ×a|zn| I in CHn+1.

Conversely, if a real hypersurface in CHn+1 is a warped product N ×f I of a

complex hypersurface N and an open interval I, then, up to rigid motions, it is

locally obtained in the way described above.

6. Warped Products as Lagrangian Submanifolds

A submanifold N in a Kahler manifold M is called a totally real submanifold

if the almost complex structure J of M carries each tangent space TxN of N

into its corresponding normal space T⊥x N [34]. The submanifold N is called a

holomorphic submanifold (or Kahler submanifold) if J carries each TxN into itself

[48].

A totally real submanifold N in a Kahler manifold M is called a Lagrangian

submanifold if dimRN = dimC M . (For the most recent survey on differential

geometry of Lagrangian submanifolds, see [17, 22].)

For Lagrangian immersions into complex Euclidean n-space Cn, a well-known

result of M. Gromov [41] states that a compact n-manifold M admits a La-

grangian immersion (not necessary isometric) into Cn if and only if the complex-

ification TM ⊗ C of the tangent bundle of M is trivial. In particular, Gromov’s

result implies that there exists no topological obstruction to Lagrangian immer-

sions for compact 3-manifolds in C3, because the tangent bundle of a 3-manifold

is always trivial.

From the Riemannian point of view, it is natural to ask the following basic

question.

Problem 6.1. When a Riemannian n-manifold admits a Lagrangian iso-

metric immersion into Cn?

Not every warped product N1 ×f N2 can be isometrically immersed in a

complex space form as a Lagrangian submanifold. However, for warped products

of curves and the unit (n − 1)-sphere Sn−1, we have the following existence

theorem.


Theorem 6.1.([13, 18]) Every simply-connected open portion of a warped

product manifold I ×f Sn−1 of an open interval I and a unit (n − 1)-sphere

admits an isometric Lagrangian immersion into Cn.

The Lagrangian immersions given in Theorem 6.1 with vanishing scalar cur-

vature have been recently determined explicitly in [5].

The Lagrangian immersions given in Theorem 6.1. are expressed in terms of

complex extensors in the sense of [13] which are defined as follows:

Let G : Mn−1 → Em be an isometric immersion of a Riemannian (n − 1)-

manifold into Euclidean m-space Em and F : I → C a unit speed curve in the

complex plane. We extend the immersion G : M n−1 → Em to an immersion of

I ×Mn−1 into complex Euclidean m-space Cm given by

φ = F ⊗G : I ×Mn−1 → C ⊗ Em = C

m, (6.1)

where F ⊗G is the tensor product immersion of F and G defined by

(F ⊗G)(s, p) = F (s) ⊗G(p), s ∈ I, p ∈Mn−1. (6.2)

Such an extension F ⊗G of the immersion G is called a complex extensor of G

(or of submanifold Mn−1) via F .

Those Lagrangian immersions obtained in Theorem 6.1 show how to embed

a time slice of the Schwarzschild space-time that models the out space around a

massive star or a black hole as Lagrangian submanifolds (cf. [5, 49]).

Since rotation hypersurfaces and real space forms can be at least locally

expressed as the warped products of curves and a unit sphere, Theorem 6.1

implies immediately the following

Corollary 6.1. Every rotation hypersurface of En+1 can be isometrically

immersed as Lagrangian submanifold in Cn.

Corollary 6.2. Every Riemannian n-manifold of constant sectional curva-

ture c can be locally isometrically immersed in Cn as a Lagrangian submanifold.

Corollary 6.3. Every rotation Every simply-connected surface equipped with

a Riemannian metric g = E2(x)(dx2+dy2) always admits a Lagrangian isometric

immersion into C2.

138 BANG-YEN CHEN

Remark 6.1. Not every Riemannian n-manifold of constant sectional curva-

ture can be globally isometrically immersed in Cn as a Lagrangian submanifold.

For instance, it is known from [15] that every compact Riemannian n-manifold

with positive sectional curvature (or positive Ricci curvature) does not admit any

Lagrangian isometric immersion into Cn.

Remark 6.2. In views of Theorem 6.1 it is interesting to ask the following.

Problem 6.2. Determine necessary and/or sufficient conditions for warped

product manifolds to admit Lagrangian isometric immersions into complex Eu-

clidean spaces or more generally into complex space forms.

7. Segre Embedding and Its Extensions

Let (zi0, . . . , z

iαi

) (1 ≤ i ≤ s) denote the homogeneous coordinates of CP αi .

Define a map:

Sα1···αs : CPα1 × · · · × CP αs → CPN , N =s

∏

i=1

(αi + 1) − 1, (7.1)

which maps a point ((z10 , . . . , z

1α1

), . . . , (zs0, . . . , z

sαs

)) of the product Kahler man-

ifold CP α1 × · · · × CP αs to the point (z1i1· · · zs

ij)1≤i1≤α1,...,1≤is≤αs in CPN . The

map Sα1···αs is a Kahler embedding which is called the Segre embedding. The

Segre embedding was constructed by C. Segre in 1891 [52].

The following results obtained in 1981 can be regarded as “converse” to Segre

embedding constructed in 1891.

Theorem 7.1.([19, 32]) Let Mα1

1 , . . . ,Mαss be Kahler manifolds of dimen-

sions α1, . . . , αs, respectively. Then every Kahler immersion

f : Mα1

1 × · · · ×Mαss → CPN , N =

s∏

i=1

(αi + 1) − 1,

of Mα1

1 ×· · ·×Mαss into CPN is locally the Segre embedding, that is, Mα1

1 , . . . ,Mαss

are open portions of CP α1 , . . . , CP αs , respectively, and moreover, the Kahler

immersion f is congruent to the Segre embedding.

This theorem was proved in [44] under two additional assumptions; namely,

s = 2 and the Kahler immersion f has parallel second fundamental form.


Let ∇kσ denote the k-th covariant derivative of the second fundamental form

for k = 0, 1, 2, · · ·. Denoted by ||∇kσ||2 the squared norm of ∇kσ.

Theorem 7.2.([19, 32]) Let Mα1

1 ×· · ·×Mαss be a product Kahler submanifold

of CPN . Then we have

||∇k−2σ||2 ≥ k! 2k∑

i1<···<ik

α1 · · ·αk, (7.2)

for k = 2, 3, · · · .The equality sign of (7.2) holds for some k if and only if M α1

1 , . . . ,Mαss

are open portions of CP α1 , . . . , CP αs , respectively, and the Kahler immersion is

congruent to the Segre embedding.

In particular, if k = 2, Theorem 7.2 reduces to

Theorem 7.3.([19, 32]) Let Mh1 ×Mp

2 be a product Kahler submanifold of

CPN . Then we have

||σ||2 ≥ 8hp. (7.3)

The equality sign of (7.3) holds if and only if M h1 and Mp

2 are open portions

of CP h and CP p, respectively, and the Kahler immersion is congruent to the

Segre embedding Sh,p.

8.CR-Products

A submanifold N in a Kahler manifold M is called a CR-submanifold [3] if

there exists on N a holomorphic distribution D whose orthogonal complement

D⊥ is a totally real distribution, i.e., JD⊥x ⊂ T⊥

x N .

A CR-submanifold of a Kahler manifold M is called a CR-product [8] if it is

a Riemannian product NK ×N⊥ of a Kahler submanifold NK and a totally real

submanifold N⊥.

For CR-products in complex space forms, the following are known.

Theorem 8.1.([8]) We have

(i) A CR-submanifold in Cm is a CR-product if and only if it is a direct sum

of a Kahler submanifold and a totally real submanifold of linear complex

subspaces.

140 BANG-YEN CHEN

(ii) There do not exist CR-products in complex hyperbolic spaces other than

Kahler submanifolds and totally real submanifolds.

CR-products NK×N⊥ in CP h+p+hp are obtained from the Segre embedding;

namely, we have the following.

Theorem 8.2.([8]) Let NhT ×Np

⊥ be the CR-product in CPm with constant

holomorphic sectional curvature 4. Then

m ≥ h+ p+ hp. (8.1)

The equality sign of (8.1) holds if and only if

(a) NhT is a totally geodesic Kahler submanifold,

(b) Np⊥ is a totally geodesic totally real submanifold, and

(c) the immersion is given by

NhT ×Np

⊥

t.g.

−−−−→CP h ×CP pShp

−−−−−−−−→Segre imbedding

CP h+p+hp. (8.2)

Theorem 8.3.([8]) Let NhT ×Np

⊥ be the CR-product in CPm with constant

holomorphic sectional curvature 4. Then the squared norm of the second funda-

mental form satisfies

||σ||2 ≥ 4hp. (8.3)

The equality sign of (8.3) holds if and only if

(a) NhT is a totally geodesic Kahler submanifold,

(b) Np⊥ is a totally geodesic totally real submanifold, and

(c) the immersion is given by

NhT ×Np

⊥

t.g.

−−−−→CP h ×CP pShp

−−−−−−−−→Segre imbedding

CP h+p+hp ⊂ CPm. (8.4)

9. Warped Products as CR-Submanifolds

For warped product CR-submanifolds in complex space forms, we propose

the following simple question:


Problem 9.1.

∀

NK ×f N⊥

or

N⊥ ×f NN

CR-immersion

−−−−−−−−→ Cm or Mm(4c) =⇒ ???

where Mm(4c) denotes a Kahler manifold of constant holomorphic sectional cur-

vature 4c.

First we mention the following.

Theorem 9.1.([19]) If N⊥×f NK is a warped product CR-submanifold of a

Kahler manifold M such that N⊥ is a totally real and NK a Kahler submanifold

of M , then it is a CR-product.

Theorem 9.1 shows that there does not exist warped productCR-submanifolds

of the form N⊥×fNK other than CR-products. So, we shall only consider warped

product CR-submanifolds of the form: NK ×f N⊥, by reversing the two factors

NK and N⊥. We simply call such CR-submanifolds CR-warped products.

CR-warped products are characterized as follows.

Theorem 9.2.([19]) A proper CR-submanifold M of a Kahler manifold M

is locally a CR-warped product if and only if

AJZX = ((JX)µ)Z, X ∈ D, Z ∈ D⊥, (9.1)

for some function µ on M satisfying Wµ = 0, W ∈ D⊥.

We have the following solution to Problem 9.1.

Theorem 9.3.([19]) Let NK ×f N⊥ be a CR-warped product in a Kahler

manifold M . Then

||σ||2 ≥ 2p ||∇(ln f)||2, (9.2)

where ∇ ln f is the gradient of ln f on NK and p = dimN⊥.

If the equality sign of (9.2) holds identically, then NK is a totally geodesic

Kahler submanifold and N⊥ is a totally umbilical totally real submanifold of M .

Moreover, NK ×f N⊥ is minimal in M .

142 BANG-YEN CHEN

When M is anti-holomorphic, i.e., when JD⊥x = T⊥

x N , and p > 1. The

equality sign of (9.2) holds identically if and only if N⊥ is a totally umbilical

submanifold of M .

If M is anti-holomorphic and p = 1, then the equality sign of (9.2) holds

identically if and only if the characteristic vector field Jξ of M is a principal

vector field with zero as its principal curvature. (Notice that in this case, M

is a real hypersurface in M .) Also, in this case, the equality sign of (9.2) holds

identically if and only if M is a minimal hypersurface in M .

Remark 9.1. Theorem 9.1 and Theorem 9.3 have been extended to twisted

product CR-submanifolds in [16].

CR-warped products in complex space forms satisfying the equality case of

(9.2) have been completely classified. In fact, we have the following.

Theorem 9.4.([19]) A CR-warped product NK ×f N⊥ in Cm satisfies

||σ||2 = 2p||∇(ln f)||2, (9.3)

if and only if

(i) NK is an open portion of a complex Euclidean h-space Ch,

(ii) N⊥ is an open portion of the unit p-sphere Sp,

(iii) there is a = (a1, . . . , ah) ∈ Sh−1 ⊂ Eh such that f =

√

〈a, z〉2 + 〈ia, z〉2 for

z = (z1, . . . , zh) ∈ Ch, w = (w0, . . . , wp) ∈ Sp ⊂ E

p+1, and

(iv) up to rigid motions, the immersion is given by

x(z, w) =(

z1 + (w0 − 1)a1

h∑

j=1

ajzj , · · · zh + a(w0 − 1)ah

h∑

j=1

ajzj ,

w1

h∑

j=1

ajzj , · · · , wp

h∑

j=1

ajzj , 0, · · · , 0)

. (9.4)

A CR-warped product NT ×fN⊥ is said to be trivial if its warping function f

is constant. A trivial CR-warped product NT ×fN⊥ is nothing but a CR-product

NT ×Nf⊥, where N f

⊥ is the manifold with metric f 2gN⊥which is homothetic to

the original metric gN⊥on N⊥.

The following result classifies CR-warped products in complex projective

spaces satisfying the equality case of (9.2)


Theorem 9.5.([20]) A non-trivial CR-warped product NT ×fN⊥ in CPm(4)

satisfies the basic equality ||σ||2 = 2p||∇(ln f)||2 if and only if we have

(1) NT is an open portion of complex Euclidean h-space Ch,

(2) N⊥ is an open portion of a unit p-sphere Sp, and

(3) up to rigid motions, the immersion x of NT ×f N⊥ into CPm(4) is the com-

position π ◦ x, where

x(z, w) =(

z0 + (w0 − 1)a0

h∑

j=0

ajzj , · · · , zh + (w0 − 1)ah

h∑

j=0

ajzj ,

w1

h∑

j=0

ajzj , · · · , wp

h∑

j=0

ajzj, 0, . . . , 0)

, (9.5)

π is the projection π : Cm+1∗ → CPm(4), a0, . . . , ah are real numbers satisfy-

ing a20 +a2

1 + · · ·+a2h = 1, z = (z0, z1, . . . , zh) ∈ Ch+1 and w = (w0, . . . , wp) ∈

Sp ⊂ Ep+1.

Similarly, we have the following classification theorems for CR-warped prod-

ucts in complex hyperbolic spaces satisfying the equality case of (9.2).

Theorem 9.6.([20]) A CR-warped product NT ×f N⊥ in CHm(−4) satisfies

the basic equality ||σ||2 = 2p||∇(ln f)||2 if and only if one of the following two

cases occurs:

(1) NT is an open portion of complex Euclidean h-space Ch, N⊥ is an open por-

tion of a unit p-sphere Sp and, up to rigid motions, the immersion is the

composition π ◦ x, where π is the projection π : Cm+1∗1 → CHm(−4) and

x(z,w) =(

z0 + a0(1−w0)h

∑

j=0

ajzj , z1 + a1(w0 − 1)h

∑

j=0

ajzj , · · · ,

zh+ah(w0−1)h

∑

j=0

ajzj , w1

h∑

j=0

ajzj , . . . , wp

h∑

j=0

ajzj , 0, . . . , 0)

, (9.6)

for some real numbers a0, . . . , ah satisfying a20 − a2

1 − · · · − a2h = −1, where

z = (z0, . . . , zh) ∈ Ch+11 and w = (w0, . . . , wp) ∈ Sp ⊂ E

p+1.

(2) p = 1, NT is an open portion of Ch and, up to rigid motions, the immersion

144 BANG-YEN CHEN

is the composition π ◦ x, where

x(z, t) =(

z0 + a0(cosh t− 1)h

∑

j=0

ajzj , z1 + a1(1 − cosh t)h

∑

j=0

ajzj ,

. . . , zh + ah(1 − cosh t)h

∑

j=0

ajzj , sinh th

∑

j=0

ajzj , 0, . . . , 0)

, (9.7)

for some real numbers a0, a1 . . . , ah+1 satisfying a20 − a2

1 − · · · − a2h = 1.

10. Another Solution to Problem 9.1

In fact, CR-warped products in a complex space form also satisfy another

general inequality; which provides the second solution to Problem 9.1.

Theorem 10.1.([26]) Let N = NhK ×f N

p⊥ be a CR-warped product in a

complex space form M(4c) of constant holomorphic sectional curvature c. Then

we have

||σ||2 ≥ 2p{||∇(ln f)||2 + ∆(ln f) + 2hc}. (10.1)

If the equality sign of (10.1) holds identically, then NK is a totally geodesic

submanifold and N⊥ is a totally umbilical submanifold. Moreover, N is a minimal

submanifold in M(4c).

One may also classifies CR-warped products which satisfy the equality case

of (10.1) as given in the following theorems.

Theorem 10.2.([26]) Let φ : NhK ×f N

p⊥ → C

m be a CR-warped product in

Cm. Then we have

||σ||2 ≥ 2p{||∇(ln f)||2 + ∆(ln f)}. (10.2)

The equality case of (10.2) holds if and only if

(a) NK is an open portion of Ch∗ =: C

h − {0};(b) N⊥ is an open portion of Sp;

(c) There is α, 1 ≤ α ≤ h, and complex Euclidean coordinates {z1, . . . , zh} on Ch

such that f =√

∑αj=1 zj zj ;


(d) Up to rigid motions, the immersion φ is given by

φ = (w0z1, . . . , wpz1, . . . , w0zα, . . . , wpzα, zα+1, . . . , zh, 0, . . . , 0), (10.3)

for z = (z1, . . . , zh) ∈ Ch∗ and w = (w0, . . . , wp) ∈ Sp ⊂ E

p+1.

Theorem 10.3.([26]) Let φ : NT ×f N⊥ → CPm(4) be a CR-warped product

with dimCNT = h and dimRN⊥ = p. Then we have

||σ||2 ≥ 2p{||∇(ln f)||2 + ∆(ln f) + 2h}. (10.4)

The CR-warped product satisfies the equality case of (10.4) if and only if

(i) NT is an open portion of complex projective h-space CP h(4);

(ii) N⊥ is an open portion of unit p-sphere Sp; and

(iii) There exists a natural number α ≤ h such that, up to rigid motions, φ is the

composition π ◦ φ, where

φ(z, w) = (w0z0, . . . , wpz0, . . . , w0zα, . . . , wpzα, zα+1, . . . , zh, 0 . . . , 0),

(10.5)

for z = (z0, . . . , zh) ∈ Ch+1∗ and w = (w0, . . . , wp) ∈ Sp ⊂ E

p+1, where π is

the projection π : Cm+1∗ → CPm(4).

Theorem 10.4.([26]) Let φ : NT ×f N⊥ → CHm(−4) be a CR-warped

product with dimC NT = h and dimRN⊥ = p. Then we have

||σ||2 ≥ 2p{||∇(ln f)||2 + ∆(ln f) − 2h}. (10.6)

The CR-warped product satisfies the equality case of (10.6) if and only if

(a) NT is an open portion of complex hyperbolic h-space CHh(−4);

(b) N⊥ is an open portion of unit p-sphere Sp (or R, when p = 1); and

(c) up to rigid motions, φ is the composition π ◦ φ, where either φ is given by

φ(z, w) = (z0, · · · , zβ , w0zβ+1, · · · , wpzβ+1, · · · , w0zh, . . . , wpzh, 0 · · · , 0),(10.7)

for 0 < β ≤ h, z = (z0, . . . , zh) ∈ Ch+1∗1 and w = (w0, . . . , wp) ∈ Sp, or φ is

given by

φ(z, u) = (z0 cosh u, z0 sinhu, z1 cos u, z1 sinu, · · · , · · · , zα cos u,

zα sinu, zα+1, · · · , zh, 0 · · · , 0), (10.8)

146 BANG-YEN CHEN

for z = (z0, . . . , zh) ∈ Ch+1∗1 and u ∈ R, where π is the projection π : C

m+1∗1 →

CHm(−4).

11. Convolution of Riemannian Manifolds

Convolution of Riemannian manifolds introduced in [23, 29] is a natural

extension of direct and warped products. Convolution of Riemannian manifolds

is defined as follows:

Let (N1, g1) and (N2, g2) be two Riemannian manifolds and f and h be

positive functions on N1 and N2, respectively. Consider the symmetric tensor

field gf,h of type (0,2) on N1 ×N2 defined by

hg1 ∗f g2 = h2g1 + f2g2 + 2fhdf ⊗ dh, (11.1)

which is called the convolution of g1 and g2 via h and f . The product manifold

N1 × N2 equipped with hg1 ∗f g2 is called a convolution manifold, denoted by

hN1F fN2.

When the scale functions f, h are irrelevant, we simply denote hN1F fN2 and

hg1 ∗f g2 by N1FN2 and g1 ∗ g2, respectively.

If hg1 ∗f g2 is nondegenerate, it defines a pseudo-Riemannian metric on

N1×N2 with index ≤ 1. In this case, hg1 ∗f g2 is called a convolution metric and

hN1FfN2 is called a convolution pseudo-Riemannian manifold. If the index of the

pseudo-Riemannian metric is zero, hN1FfN2 is called a convolution Riemannian

manifold.

The notion of convolution arises naturally, since each tensor product immer-

sion gives rise to a convolution as shown in the next theorem.

Theorem 11.1.([23]) Let ψ1 : (N1, g1) → En∗ ⊂ E

n and ψ2 : (N2, g2) →E

m∗ ⊂ E

m be isometric immersions, where En∗ = E

n − {0}. Then

ψ = ψ1 ⊗ ψ2 : N1 ×N2 → En ⊗ E

m = Enm; (u, v) 7→ ψ1(u) ⊗ ψ2(v), (11.2)

gives rise to a convolution manifold N1FN2 equipped with

ρ2g1 ∗ρ1

g2 = ρ22g1 + ρ2

1g2 + 2ρ1ρ2dρ1 ⊗ dρ2, (11.3)

where ρ1 and ρ2 are the distance functions of ψ1 and ψ2, respectively.


Therefore, we have the following induced metrics:

φ1 ⊕ φ2 : N1 ×N2 → Em1 ⊕ E

m2 =⇒ g1 + g2 (direct sum)

ψ1 ⊗ ψ2 : N1 ×N2 → En ⊗ E

m =⇒ ρ2g1 ∗ρ1

g2 (convolution)

Example 11.1. If ψ2 : (N2, g2) → Em is an isometric immersion such that

ψ2(N2) is contained in Sm−1(1) ⊂ Em which is centered at the origin. Then the

convolution ρ2g1 ∗ρ1

g2 on N1FN2 is the warped product metric: g1 + ρ21g2.

Definition 11.1. A convolution hg1 ∗ fg2 of two Riemannian metrics g1 and

g2 is called degenerate if det(hg1 ∗f g2) = 0 holds identically.

For X ∈ T (Nj), j = 1, 2, we denote by |X|j the length of X with respect to

metric gj on Nj .

The following theorem determines when a convolution is degenerate.

Theorem 11.2.([23]) Let hN1F fN2 be the convolution of Riemannian man-

ifolds (N1, g1) and (N2, g2) via h and f . Then hg1 ∗f g2 is degenerate if and only

if we have

(1) the length |grad f |1 of the gradient of f on (N1, g1) is a nonzero constant, say

c, and

(2) the length |grad h|2 of the gradient of h on (N2, g2) is the constant given by

1/c, i.e., the reciprocal of c.

The following result provides a necessary and sufficient condition for hg1∗f g2

to be Riemannian.

Theorem 11.3.([23]) Let hN1F fN2 be the convolution of Riemannian man-

ifolds (N1, g1) and (N2, g2) via h and f . Then hg1 ∗ fg2 is Riemannian if and

only if we have |grad f |1 |gradh|2 < 1.

In views of Theorems 11.2 and 11.3, it is natural to study Euclidean sub-

manifolds whose distance function ρ satisfies the condition: |grad ρ| = constant.

To do so, we introduce the notion of constant-ratio submanifolds.

Definition 11.2. Let x : M → En be an isometric immersion. We put

x = xN + xT , (11.4)

148 BANG-YEN CHEN

where xN and xT are the normal and tangential components of the position vector

field, respectively. The submanifold M is said to be of constant-ratio if the ratio

|xN | : |xT | is constant; or equivalently, |xT | : |x| = c = constant (cf. [21]).

The following results characterize constant-ratio submanifolds.

Theorem 11.4.([23]) Let x : M → En be an isometric immersion. Then

M is of constant-ratio with |xT | : |x| = c if and only if |grad ρ| = c which is

constant.

In particular, for submanifold of constant-ratio, we have |grad ρ| = c ≤ 1.

Theorem 11.3 and Theorem 11.4 imply

Corollary 11.1. Let ψ1 : (N1, g1) → En and ψ2 : (N2, g2) → E

m be isometric

immersions and ρ1 = |ψ1| and ρ2 = |ψ2| be their distance functions. Then

ρ2g1 ∗ρ1

g2 is degenerate if and only if |grad ρ1| = |grad ρ2| = 1.

Space-like constant-ratio submanifolds in pseudo-Euclidean space have been

completely classified in [25]. In particular, for constant-ratio submanifolds in

Euclidean space, we have the following.

Theorem 11.5.([25]) An isometric immersion x : M → Em is of constant-

ratio if and only if one of the following three cases occurs:

(1) x(M) is contained in hypercone with vertex at the origin.

(2) x(M) is contained in a hypersphere Sm−1(r2) of Em centered at the origin.

(3) There is a number b ∈ (0, 1) and local coordinate system {s, u2, . . . , un} on

M such that the immersion x is given by

x(s, u2, . . . , un) = bs Y (s, u2, . . . , un), (11.5)

where Y = Y (s, u2, . . . , un) satisfies conditions:

(3.a) Y = Y (s, u2, . . . , un) lies in Sm−1(1),

(3.b) Ys is perpendicular to Yu2, . . . , Yun , and

(3.c) |Ys| =√

1 − b2/(bs).

Remark 11.1. Submanifolds of constant-ratio also relate to a problem in

physics concerning the motion in a central force field which obeys the inverse-

cube law. For example, the trajectory of a particle subject to a central force of


attraction located at the origin which obeys the inverse-cube law is a curve of

constant-ratio.

Remark 11.2. The inverse-cube law was originated from Sir Isaac Newton

in his letter sent to Robert Hooke on December 13, 1679. This letter is of great

historical importance because it reveals the state of Newton’s development of

dynamics at that time (cf. [45, 46]).

12. Applications of Convolution

We define an isometric map from a convolution manifold into a Riemannian

manifold as follows.

Definition 12.1. A map ψ : (hN1FfN2, hg1 ∗ fg2) → (M, g) from a convo-

lution manifold into a Riemannian manifold is called isometric if ψ∗g = hg1 ∗ fg2.

For example, let z : Ch∗ → C

h and x : Ep∗ → E

p be inclusion maps. Consider

the map

ψz,x = z ⊗ x : Ch∗ × E

p∗ → (Chp, g0), (12.1)

defined by

ψz,x = z ⊗ x = (z1x1, . . . , z1xp, . . . , zhx1, . . . , zhxp), (12.2)

for z = (z1, . . . , zh) ∈ Ch∗ and x = (x1, . . . , xp) ∈ E

p∗. Then we have

ψ∗z,x(g0) = ρ2

g1 ∗ρ1g2 = ρ2

2g1 + ρ21g2 + 2ρ1ρ2dρ1 ⊗ dρ2, (12.3)

where g1, g2 and ρ1, ρ2 are the metrics and distance functions of Ch∗ ,E

p∗, respec-

tively. Thus

ψz,x : (Ch∗ × E

p∗, ρ2

g1 ∗ρ1g2) → (Chp, g0) (12.4)

is an isometric map.

An isometric map

φ : (Ch∗ × E

p∗, ρ2

g1 ∗ρ1g2) → (Cm, g0) (12.5)

is call a CR-map if φ carries each slice Ch∗ × {v} of C

h∗ × E

p∗ into a Kahler sub-

manifold and each slice {u} × Ep∗ of C

h∗ × E

p∗ into a totally real submanifold.

150 BANG-YEN CHEN

The notion of convolution is useful in the study of tensor product immersions.

For instance, we may applying the notion of convolution to obtain the following

two simple characterizations of the map ψz,x = z⊗ x : Ch∗ × E

p∗ → C

hp defined by

(12.2).

Theorem 12.1.([29]) Let φ : U → Cm be an isometric CR-map from an

open portion U of the convolution manifold (Ch∗ × E

p∗, ρ2

g1 ∗ρ1g2) into complex

Euclidean m-space. Then we have

(1) m ≥ hp.

(2) If m = hp, then, up to rigid motions of Cm, φ is the map ψz,x = z⊗x defined

by (12.2).

Theorem 12.2.([29]) Let φ : U → Cm be an isometric CR-map from an

open portion U of the convolution manifold (Ch∗ × E

p∗, ρ2

g1 ∗ρ1g2) into a complex

Euclidean space. Then the second fundamental form σ of φ satisfies

||σ||2 ≥ (2h − 1)(p− 1)

ρ21ρ

22

. (12.6)

The equality sign of (12.6) holds identically if and only if, up to rigid motions

of Cm, φ is given by φ(z, x) = (z ⊗ x, 0, . . . , 0).

These two theorems can be regarded as Euclidean version of Theorems 7.1

(with s = 2) and 7.3 related with the converse to Segre imbedding.

13. T -Constant and N-Constant Submanifolds

In views of constant-ratio submanifolds in Euclidean space, it is natural to

study submanifolds in Euclidean space such that either xT or xN has constant

length, where xT and xN are the tangential and the normal components of the

position function given in (11.4). For simplicity, we call a submanifold M in a

Euclidean space a T -constant (respectively, N -constant) submanifold if xT is of

constant length (respectively, |xN | is of constant length).

For T -constant submanifolds, we have the following.

Theorem 13.1.([30]) Let x : M → Em be an isometric immersion of a

Riemannian n-manifold into the Euclidean m-space. Then M is a T -constant

submanifold if and only if either


(i) x(M) is contained in a hypersphere Sm−1(r2) of Em, or

(ii) there exist a real numbers a, b and local coordinate systems {s, u2, . . . , un}on M such that the immersion x is given by

x(s, u2, . . . , un) =√

a2 + b+ 2as Y (s, u2, . . . , un), (13.1)


(2.a) Y = Y (s, u2, . . . , un) lies in the unit hypersphere Sm−1(1),

(2.b) the coordinate vector field Ys is perpendicular to coordinate vector fields

Yu2, . . . , Yun , and

(2.c) Ys satisfies |Ys| =√b+ 2as/(a2 + b+ 2as).

For N -constant submanifolds, we have the following.

Theorem 13.2.([30]) Let x : M → Em be an isometric immersion of a

Riemannian n-manifold into the Euclidean m-space. Then M is N -constant if

and only if one of the following three cases occurs:

(1) x is a conic submanifold with the vertex at the origin.

(2) x(M) is contained in a hypersphere Sm−1(r2) of Em.

(3) There exist a positive number c and local coordinate systems {s, u2, . . . , un}on M such that the immersion x is given by

x(s, u2, . . . , un) =√

s2 + c2 Y (s, u2, . . . , un), (13.2)


(3.a) Y = Y (s, u2, . . . , un) lies in the unit hypersphere Sm−1(1),

(3.b) the coordinate vector field Ys is perpendicular to coordinate vector fields

Yu2, . . . , Yun , and

(3.c) Ys satisfies |Ys| = c/(s2 + c2).

14. Rectifying curves and τ/κ

Consider a unit-speed space curve x : I → R3, I = (α, β), in E

3 that has at

least four continuous derivatives. Let t denote x′. It is possible that t′(s) = 0

for some s, however, we assume that this never happens. Then we can introduce

a unique vector field n and positive function κ so that t′ = κn. We call t′ the

152 BANG-YEN CHEN

curvature vector field, n the principal normal vector field, and κ the curvature.

Since t is a constant length vector field, n is orthogonal to t. The binormal vector

field is defined by b = t×n which is a unit vector field orthogonal to both t and

n. One defines the torsion τ by the equation b′ = −τn.

The famous Serret-Frenet equations are given by

t′ = κn,

n′ = −κt + τb, (14.1)

b′ = −τn.

At each point of the curve, the planes spanned by {t,n}, {t,b} and {n,b}are known as the osculating plane, the rectifying plane and the normal plane,

respectively. A curve in E3 is called a twisted curve if has nonzero curvature and

torsion.

In elementary differential geometry it is well-known that a curve in E3 is a

planar curve if its position vector lies in its osculating plane at each point. And

it is a spherical curve if its position vector lies in its normal plane at each point.

In views of these basic facts, it is natural to ask the following simple geometric

question:

Problem 14.1. When the position vector of a space curve x : I → R3 always

lies in its rectifying plane ?

For simplicity we call such a curve a rectifying curve. So, for a rectifying

curve x : I → E3, the position vector x satisfies

x(s) = λ(s)t(s) + µ(s)b(s), (14.2)

for some functions λ and µ.

The following result provides some simple characterizations of rectifying

curves. In particular, it shows that rectifying curves in E3 are N -constant curves.

Theorem 14.1.([31]) Let x : I → E3 be a rectifying curve in E

3 with κ > 0

and s be its arclength function. Then we have

(i) The curve is N -constant and the distance function ρ is non-constant.


(ii) The distance function ρ = |x| satisfies ρ2 = s2 + c1s+ c2 for some constants

c1 and c2.

(iii) The tangential component of the position vector of the curve is given by

〈x, t〉 = s+ c for some constant c.

(iv) the torsion τ is non-zero and the binormal component of the position vector

is constant, i.e. 〈x,b〉 is constant.

Conversely, if x : I → E3 is a curve with κ > 0 and one of (i), (ii), (iii) or

(iv) holds, then x is a rectifying curve.

The fundamental existence and uniqueness theorem for curves in E3 states

that a curve is uniquely determined up to rigid motions by its curvature and

torsion, given as functions of its arclength. Furthermore, given continuous func-

tions κ(s) and τ(s), with κ(s) positive and continuously differentiable, there is

a differentiable curve (of class at least C3) with curvature κ and torsion τ . In

general, a result of Lie and Darboux shows that solving Serret-Frenet equations

is equivalent to solving a certain complex Riccati equation (see [53, p.36]). In

practice, space curves with prescribed curvature and torsion are often impossible

to find explicitly. Fortunately, we are able to determine completely all rectifying

curves in E3.

Theorem 14.2.([31]) Let x : I → E3 be a curve E

3 with κ > 0. Then it is a

rectifying curve if and only if, up to parameterization, it is given by

x(t) = a(sec t)y(t), (14.3)

where a is a positive number and y = y(t) is a unit speed curve in S2, where S2

denotes the unit sphere in E3 centered at the origin.

A generalized helix in Euclidean 3-space is a twisted curve whose unit tangent

vector field t makes constant angle with a fixed direction. It is well-known that

a twisted curve is a generalized helix if and only if the ratio τ/κ is a non-zero

constant. So, it is natural to ask the following.

Problem 14.2. When the ratio τ/κ of a twisted curve is a non-constant

linear function in arclength function ?

The following result provides an answer to this Problem.

154 BANG-YEN CHEN

Theorem 14.3.([31]) Let x : I → E3 be a twisted curve in E

3. Then x is

congruent to a rectifying curve if and only if the ratio of torsion and curvature

of the curve is a non-constant linear function in arclength function s, i.e., τ/κ =

c1s+ c2 for some constants c1, c2 with c1 6= 0.

Theorem 14.3 seems to be of independent interest by itself.

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Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA.

E-mail: [email protected]

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