COMPACT SUBMANIFOLDS OF CODIMENSION 3 - California State
Transcript of 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 Manifolds Classification Homotopy Applications Bibliography
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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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)).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Manifolds
Manifolds
Figure: Manifold: an easier definition
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Manifolds
Figure: The (green) tangent space of the (orange) manifold is the set of(yellow) tangent vectors of (black) curves in the point p.
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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 .
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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)
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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
.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
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∧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 .
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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 .
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Classification theorem
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 .
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Classification theorem
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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Classification theorem
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).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Classification theorem
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).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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Classification theorem
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)
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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 .
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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).
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3
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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.
MARTINS, R. M, MEDEIROS, R. A, RICHELSON, S. COMPACT SUBMANIFOLDS OF CODIMENSION 3