COMPACT SUBMANIFOLDS OF CODIMENSION 3 - California State

Introduction Manifolds Classification Homotopy Applications Bibliography

INTERNATIONAL RESEARCH EXPERIENCE FOR STUDENTSUNIVERSIDADE ESTADUAL DE CAMPINAS - UNICAMP

ON THE COMPACT SUBMANIFOLDS OF CODIMENSION 3WITH NONNEGATIVE CURVATURE

MARTINS, R. M., MEDEIROS, R. A., RICHELSON, S.Advisor: Maria Helena Noronha

Support: CNPq/NSFCAMPINAS, SP - BRAZIL

JULY/2006

MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3


Introduction

In this project, we’ve studied the holonomy algebra of a manifoldimmersed in Rn+3.

Also, we applied our classification result in a problem related to coveringspaces of a compact manifold with non-negative curvature.



Vector fields, connections

Vectors fields

We will denote by RNp the set of vectors that have origin at p.

Definition

A vector field in RN is a map X : RN → RNp .

Identifying the vectors {ei}Ni=1 of the standard basis of RN with their

translates {eip}Ni=1 at the point p, we write

X (q) =N∑

i=1

ai (q)ei .

A vector field X is said to be differentiable if the functions ai ’s aredifferentiable.



Vector fields, connections

Covariant derivatives and connections

Definition

Let X be a differentiable vector field in RN and c : [0, 1] → RN adifferentiable curve in RN . Setting X (c(t)) = (x1(t), . . . , xN(t)), thecovariant derivative of X over c(t) is given by

DX

dt= (x ′1(t), . . . , x

′N(t)).

Definition

Given two vector fields X and Y in RN , we define the connection ∇XYas the vector field that at p ∈ RN is given by the following procedure:Let c(t) be a curve such that c(0) = p and c ′(0) = X (p). Consider Ydefined over c(t), Y (c(t)) = (y1(t), . . . , yN(t)). Then,

∇XY (p) =DY

dt(0) = (y ′1(t), . . . , y

′N(t)).



Manifolds

Manifolds

Definition

Let M be an topological Hausdorff space with a countable basis. ThenM is said to be a manifold if there existes a natural number n and foreach point p ∈M an open neighbourhood U of p and a continuous mapφ : U → Rn which is a homeomorphism onto its image φ(U), an opensubset of Rn. The pair (U, φ) is called a chart on M. The naturalnumber N is called the dimension of M. To denote that the dimensionof M is n we write Mn.



Manifolds

Manifolds

Figure: Manifold: an easier definition



Manifolds

Immersions and tangent space

Definition

We say that a map f : Mn → RN is an immersion if for every p ∈M,exists an open set W ⊆M (p ∈ W ) and a map φ : W → U ⊆ Rn, Uopen in Rn, such that the jacobian matrix of f ◦ φ−1 : U → RN has rankn.

For a differentiable curve α : (−ε, ε) →M such that α(0) = p, thevector α′(t) is said to be a tangent vector to M in the point p.

Definition

For a point p ∈M, we define the tangent space TpM as follows:

TpM = {α′(0) | α : (−ε, ε) →M, α(0) = p}.

The number N − n is called codimension of the immersion.



Manifolds

Figure: The (green) tangent space of the (orange) manifold is the set of(yellow) tangent vectors of (black) curves in the point p.



Manifolds

Definition

A (tangent) vector field in M is a map X that associates to each pointp ∈M a vector X (p) ∈ TpM.

Given two vector fields on M, X and Y , we can extend them to X andY vector fields in RN , then define a connection in M as

∇XY (p) = (∇X Y (p))T ,

where (·)T denotes the orthogonal projection onto TpM. So, we have:

∇XY = ∇XY + (∇X Y )N ,

with (·)N the orthogonal projection onto T⊥p M.

We denote (∇X Y )N by α(X ,Y ) and call it the second fundamentalform of the immersion. Note that α is a symmetric bilinear form.



Normal field and shape operators

Definition

A normal vector field in M is a map ξ that associates to each pointp ∈M a vector ξ(p) ∈ T⊥

p M.

Given a tangent vector field in M and a normal vector field ξ, we definethe normal connection as

∇⊥X ξ = (∇X ξ)N .

The tangent component of ∇xξ defines an operator

Aξ : TpM→ TpM,

called shape operator (or Weingarten operator), given by

Aξ(X ) = −(∇X ξ)T .



Normal field and shape operators

Some identities

The shape operator is symmetric. Now we’ll present some importantidentities.

The Gauss formula:

∇XY = ∇XY + α(X ,Y ) (1)

〈AξX ,Y 〉 = 〈X ,AξY 〉 = 〈ξ, α(X ,Y )〉 (2)



Curvature tensor

Curvature tensor

Definition

Let χ(M) denote the set of vectors fields in M. For X ,Y ,Z ∈ χ(M)we define the curvature tensor as

R(X ,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X ,Y ]Z ,

and the sectional curvature of the plane spanned by X and Y as

k(X ,Y ) =〈R(X ,Y )Y ,X 〉||X ∧ Y ||2

.



Curvature tensor

Curvature tensor II

Using the equations 1 and 2, we can express the curvature tensor and thesectional curvature as

R(X ,Y )Z = Aα(Y ,Z)X − Aα(X ,Z)Y (3)

k(X ,Y ) = 〈α(X ,X ), α(Y ,Y )〉 − 〈α(X ,Y ), α(X ,Y )〉2 (4)

The equation 3 is know as Gauss Equation, and it implies the equation 4.



Curvature tensor

Fundamental theorem of submanifolds

Let

(DXα)(Y ,Z ) = ∇⊥X α(Y ,Z )− α(∇XY ,Z )− α(Y ,∇XZ )

(DY α)(X ,Z ) = ∇⊥Y α(X ,Y )− α(∇Y X ,Z )− α(X ,∇Y Z ).

Codazzi Equation:

(DXα)(Y ,Z ) = (DY α)(X ,Z ) (5)

Ricci Equation:

〈R⊥(X ,Y )ξ1, ξ2〉 = 〈[Aξ1 ,Aξ2 ](X ),Y 〉 (6)

Theorem

If Md is a d-dimensional manifold, then M may be isometrically andlocally embedded in a D-dimensional space MD if and only if its metric,twisting vector and the extrinsic curvature satisfy the Gauss, Codazzi andRicci equations.



Exterior algebra

Exterior algebra

Let V be an F-vector space. Given two vectors u, v ∈ V, we defineu ∧ v : V × V → F as

(u ∧ v)(a, b) = u∗(a)v∗(b)− u∗(b)v∗(a),

for a, b ∈ V, where u∗ and v∗ are the functionals associated with u and v .

Definition

The exterior algebra of V, denoted by∧2(V), is the set of all possible

wedge products u ∧ v , for u, v ∈ V.



∧2(V ) Lie Algebra

Exterior algebra and o(V)

The exterior algebra is a F-vector space, with the usual sum and scalarproduct. Moreover,

∧2(V) is a Lie algebra, with the bracket defined by

[x ∧ y , z ∧ w ] = 〈x , z〉y ∧ w + 〈y ,w〉x ∧ z − 〈x ,w〉y ∧ z − 〈y , z〉x ∧ w ,

for x ∧ y , z ∧ w ∈∧2(V)

We can identify∧2(V) with o(V), the algebra of skew-symmetric

operators on V in this way: for x ∧ y ∈∧2(V) we associate the operator

(x ∧ y) : V → V such that, for z ∈ V, (x ∧ y)(z) = 〈x , z〉y − 〈y , z〉x .



Curvature operator

Curvature operator

We’ll start the transition from geometry to algebra.

Definition

Let M be a manifold and p ∈M. We define the curvature operator asthe operator R :

∧2(TpM) →∧2(TpM) implicit given by

〈R(X ,Y )Y ,X 〉 = 〈R(X ∧ Y ),X ∧ Y 〉.

The curvature operator in p may be also given as

R(X ∧ Y ) =∑

ξ

Aξ(X ) ∧ Aξ(Y ), (7)

for ξ the normal fields in p (in other words, Aξ are the components of2nd fundamental form). As our spaces are finite-dimensional, we adoptthe notation {Si}i for {Aξi}i .



Curvature operator

Some definitions

Given a symmetric operator S we denote by D(S) the subspace which issimultaneously the range of S , the orthogonal complement of the kernelof S , and the span of the nonnullity eigenvectors of S .

For a immersed manifold M, if p is its codimension, the relativecurvature subspace at m is , k(m) = D(S1) + . . . + D(Sp).

The Lie group consisting of parallel translations of M around loops basedat m is called the holonomy group at m. The holonomy groups atdifferent points are isomorphic. We shall denote its Lie algebra hm. Infact, hm is spanned by the parallel translates to m of the curvaturetransformations at all of the points of M.

Denote by r(m) the Lie algebra generated by the curvaturetransformations at m; thus r(m) is a subalgebra of hm and it is clear that

r(m) ⊆∧2(k(m)) for every m.



Classification theorem

We will present a (partial) classification theorem for r(m) of immersedmanifolds with codimension 3.

Just to fix our notations, let S1, S2, S3 be the components of the 2ndfundamental form of M at the point m and:

R(x ∧ y) = S1(x) ∧ S1(y) + S2(x) ∧ S2(y) + S3(x) ∧ S3(y),

V1 = D(S1), V2 = D(S2), V3 = D(S3),

k(m) = V = V1 + V2 + V3,

r(m)=algebra generated by Im(R).




Equation ?

Our manifolds satisfies the following relations: if Zi are the sets definedby

Zi = {v ∈ V | v ∈ Vj , j 6= i},then we assume that

Zi 6= 0, i = 1, 2, 3. (8)

Figure: We need one element in Vi which is not on any other Vj .




Some lemmas

The classification for r(m) will be given after some lemmas. We will workwith the cases V 6= V1 and S1 has nonzero signature.

Lemma∧2(V) is generated by V1 ∧ V2 + V1 ∧ V3.

Lemma

If v 6= 0 is a vector in V, then v ∧ V generates∧2(V).

Lemma

If v1 ∈ V1, v2 ∈ V2 and v3 ∈ V3, all of them nonzero with V1 ∩ V2 6= 0and V1 ∩ V3 6= 0, then

∧2(V) is generated by v1 ∧ V1 + v2 ∧ V2 + v3 ∧ V3.




Some lemmas

Lemma

If vi ∈ Vi , i = 1, 2, 3 are vectors such that v1 ∧ (v2 + v3) 6= 0 then∧2(V)

is generated by∧2(V1) +

∧2(V2) +∧2(V3) + v1 ∧ v2 + v1 ∧ v3.

Lemma

If Vi ⊥ Vj for every i , j = 1, 2, 3, if dim V1 or dim (V2 + V3) do notequals 2, and if dim V2 or dim V3 do not equals 2, thenX =

∧2(V1) +∧2(V2) +

∧2(V3) is a proper subalgebra of∧2(V).

Lemma

If V1 ∩ (V2 + V3) = 0, V2 ∩ V3 = 0, V1 6= (V2 + V3)⊥, V2 * V⊥3 and

dim V1 > 2, dim V2 > 2, dim V3 > 1, then∧2(V) is generated by∧2(V1) +

∧2(V2) +∧2(V3).




Somme lemmas

Lemma

The range of R contains

S1(V2 + V3)⊥ ∧ V1 + S2(V1 + V3)

⊥ ∧ V2 + S3(V1 + V2)⊥ ∧ V3.

If Vi

⋂(Vj + Vk) for i , k 6= i , then the range of R equals

2∧(V1) +

2∧(V2) +

2∧(V3).

Lemma

The subalgebra generated by∧2(V1) +

∧2(V2) +∧2(V3) is r(m).




The theorem

Theorem

Let M be a manifold, m ∈M and r(m) the Lie algebra generated by theimage of curvature operator R : TmM→ TmM, as defined in 7. Thereare only the following options for r(m):

r(m) =

o(n)o(k)⊕ u(2)o(k)⊕ o(n − k)o(k1)⊕ o(k2)⊕ o(n − k1 − k2)



Covering spaces

Covering spaces

Definition

Let M be a manifold. We say that a pair (E , p), p : E →M, is acovering space of M if for every m ∈M has an open set m ∈ U ⊆Msuch that p−1(U) is a disjoint union of open sets ωi in E , each of whichis mapped homeomorphically onto U by p.

Definition

We say that a manifold M has nonnegative sectional curvature if forevery point p ∈M and every plane {X ,Y }, where X and Y are vectorfields in TpM, we have k(X ,Y ) ≥ 0.



Homotopy

Homotopy

Let σ : [0, 1] → X and τ : [0, 1] → X be paths in a topological space Xwith the same endpoints, σ(0) = τ(0) = x0, σ(1) = τ(1) = x1.

We say that σ and τ are homotopic if there is a continuous mapF : [0, 1]× [0, 1] → X such thati) F (s, 0) = σ(s) ii) F (s, 1) = τ(s)iii) F (0, t) = x0 iv) F (1, t) = x1



Fundamental group

Fundamental group

Consider now a topological space X and x0 ∈ X . If c is a loop at x0, let[c] denote the equivalence class of all loops at x0 that are homotopic to c .

Let Π1(X , x0) be the group of all homotopy classes of loops at x0, withthe composition operation.

Theorem

If X is path connected, then for all x , y ∈ X, Π1(X , x) ∼= Π1(X , y).



Fundamental group

Homotopy

Thanks to uniqueness (up to isomorphism), we can define thefundamental group of X and denote it by Π1(X ).

Definition

We say that a space X is simply connected if Π1(X ) = {1}, that is,every loop is homotopically trivial.

Theorem

Every connected manifold has a covering that is simply connected. Suchspace is unique (up to diffeomorphisms) and is called universal covering.

Theorem

Let M be a compact manifold with a compact universal covering. ThenΠ1(M) is finite.



Cheeger-Gronoll and consequences

Cheeger-Gronoll

Theorem (Cheeger-Gronoll)

Let M be a compact manifold such that k ≥ 0. Then the universalcovering M is given by

M = M × Rl ,

where M is a compact manifold.

Let M be a compact manifold, k ≥ 0, (M, p) a covering space and f animmersion.

M →p Mn →f Rn+3

Then f = f ◦ p : M → Rn+3 is a immersion.



Cheeger-Gronoll and consequences

Theorem

Exists x ∈M such that dim N(x) = 0, where

N(x) = {X ∈ TpM | α(X ,Y ) = 0, ∀ Y ∈ TpM}.

So, for f , there exists x ∈ M such that dim N(x) = 0 or, in other words,dim V = n.

In this case, we can apply our classification theorem.



Our application

As we have proved, the options for r(x) are

r(x) =

o(n)o(k)⊕ u(2)o(k)⊕ o(n − k)o(k1)⊕ o(k2)⊕ o(n − k1 − k2)

But by Cheeger-Gronoll theorem we know that r(x) ⊆ o(n − l)⊕ o(l).Let us analyse the options for l .



Our application

If l = 0, thenM = M × R0.

So M and M are compact and Π(M) is finite.

If l = 1 then we have the following options:

if n = 5, then o(n − l) becames o(4), and u(2) ⊆ o(4). So, r(x)may be u(2).the other options are

o(1)⊕ o(n − 1)o(1)⊕ o(k)⊕ o(n − 1− k)

If l = 2 we have r(x) ⊆ o(n − 2). The only options where o(n − 2)appears is when r(x) = o(1)⊕ o(1)⊕ o(n − 2).



Our application

For l ≥ 3, our classification theorem says that we can’t have nothingnewer. So, we have the following:

Theorem

Let Mn a compact manifold immersed in Rn+3, with k ≥ 0 and M as acovering space. Then, either Π1(M) is finite or

M = Mn−l × Rl ,

for l ≤ 2.



Bibliography

R. L. Bishop,The holonomy algebra of immersed manifolds of codimension twoJournal of Differential Geometry, Vol. 2, pp. 347-353, 1968.

M. P. do Carmo,Riemannian GeometryBirkhauser, 1992.

M. Dajczer,Submanifolds and Isometric Immersions,Mathematics Lecture Notes Series # 13, Inc. Houston, 1990.

S. Gudmundsson,An Introduction to Riemannian Geometry,Lecture Notes,http://www.matematik.lu.se/matematiklu/personal/sigma/index.html.



Acknowledgments

The research done on this article was performed during July 2006 whileparticipating in IRES at IMECC/Unicamp, funded by National ScienceFoundation (USA) and CNPq (Brasil).

The authors of the paper would like to thanks our advisor Maria HelenaNoronha and the rest of the mathematics department at the StateUniversity of Campinas for their hospitality and valuable help in thisproject. Specially, we also wish to thank Helena Lopes for theorganization of the event.


COMPACT SUBMANIFOLDS OF CODIMENSION 3 - California State

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