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Home , Differentiable curve, Differentiable manifold

Differential

Differentiable

Definition of topological : It is a topological (E, τ) so that

1. It is Hausdorff. 2. ∀x ∈ E there exists (U, ϕ) with U open and x ∈ U, such that ϕ : U → ϕ(U) is a . The pair (U, ϕ) is called chart and the real numbers (x1, . . . , xn) = ϕ(x) are called local coordinates. 3. (E, τ) has a countable of open sets.

Remark. Topological Manifolds are paracompact, i.e, every open cover has a locally finite refinement.

Remark. Paracompactness implies having .

Definition. A differentiable [C∞,Ck,Cω] structure on a topological ma- nifold M is a family of charts U = {(Uα, ϕα} so that S 1. Uα = M −1 2. If Uα ∩ Uβ 6= ∅ then ϕβ ◦ ϕα : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ) are differentiable [C∞,Ck,Cω] with differentiable [C∞,Ck,Cω] inverse. In this case we say that (Uα, ϕα) and (Uβ, ϕβ) are compatible. 3. Completness property: If (V, ψ) is a chart which is compatible with every (Uα, ϕα) ∈ U then (V, ψ) ∈ U. Remark.It is not necessary to verify the third property because of the following proposition.

Proposition. Let M be Hausdorff with countable basis of open sets. Let {(Vα, ψα): α ∈ A} be a covering of M by C∞-compatible coordinate charts. Then ∃! C∞ structure containing the charts {(Vα, ψα): α ∈ A}. Examples of topological manifolds:

1. (Rn, Id) 2. If M is a C∞ manifold and N ⊂ M is an open subset then N is a C∞ manifold too.

1 3. By the previous example GL(n, R) is a manifold. 4. If M and N are C∞ manifolds so is M × N. 5. S2 is a Cω manifold.

X Definition of open equivalence relation: Let π : X → ∼ be the projection S −1 and [U] := x∈U {π (x)}. We will say that ∼ is open iff [U] is open for every U open.

Proposition.

1. ∼ is open iff π : X → X/ ∼ is open. 2. If ∼ is open and X has a countable basis of open sets, then so does X/ ∼.

Definition. Let ∼ be an equiv. relation on X. The graph R of ∼ is the subset of X × X defined by R = {(x, y): x ∼ y}.

Proposition. Let ∼ be an open eq. relation on a Hausdorff space. Then X/ ∼ is Hausdorff iff R is closed in X × X.

Applications.

n+1 1. Pn(R) is a manifold because ∼ is open and its graph is closed .(Recall, R −0 is endowed with the equivalence relation x ∼ y iff ∃t 6= 0 so that x = ty. n−1 R −0 Pn(R) := ∼ .) 2. G(n, k) is a C∞ manifold of k(n−k). ("G(n, k) = {k-dim’l subspaces n F (n,k) kn of R }", actually G(n, k) := ∼ where F (k, n) is a manifold in R of k- frames in Rn).

Definition. Let M be a manifold and W ⊂ M open. We will say that f : W → R is a differentiable [C∞,Ck,Cω] map if ∀x ∈ W ∃(U, ϕ) coordinate chart (with x ∈ U) so that f ◦ ϕ−1 : ϕ(W ∩ U) → R is C∞.

Definition. Let A ⊂ M. We say that f : A → R is C∞ if it has a C∞ extension to an open U ⊂ M such that A ⊂ U.

Definition. Let M and N be differentiable [C∞,Ck,Cω] manifolds. We will say that f : M → N is a differentiable [C∞,Ck,Cω] map if ∀x ∈ M ∃(U, ϕ) coordinate chart with x ∈ U and (V, ψ) coordinate with f(x) ∈ V so that f(U) ⊂ V and ψ ◦f ◦ϕ−1 : ϕ(W ∩U) → R is differentiable [C∞,Ck,Cω] as a map of euclidean spaces.

2 Definition. Let M and N be C∞ manifolds and f : M → N be a C∞ map. We define the of f as the rank of ψ ◦ f ◦ ϕ−1 where (U, ϕ) and (V, ψ) are charts as h ∂yi i −1 1 n above. That is, (rank f)x := rank ∂xj |ϕ(x) where ψ ◦ f ◦ ϕ = (y , . . . , y ). Definition. Let f, g be C∞ in a neighb. of x ∈ M. We will say that f ∼ g if ∃U open so that f(y) = g(y) ∀y ∈ U. The class [f] is called of C∞ function at x. The set of germs at x will be denoted by C∞(x, R).

Definition. Let x ∈ M and X : C∞(x, R) → R. If for every chart (U, ϕ) about x we have that there exist a1, . . . , an ∈ R so that

n X ∂ −1 X([f]) = ai (f ◦ ϕ ) , ∂x ϕ(x) i=1 i we will say that X is a vector at x . Observation. If the equation holds for some chart (U, ϕ) about x, then it holds for every C∞-compatible chart overlapping at x.

Properties of tangent vectors:

1. X([f] + [g]) = X([f]) + X([g]) 2. X([λf]) = λX([f]) ∀λ ∈ R 3. X([f][g]) = X([f])[g] + [f]X([g])

Definition. The to M at x is the of tangent vectors based at x. We will denote it by TxM.

Remark. If n = dimM then dimTxM = n.

 ∂ Definition. Given a coordinate chart (U, ϕ) about x ∈ M the basis ∂xi x , 1 ≤ i ≤ n is called the natural basis of TxM associated to this chart. ∂ ∂f◦ϕ−1 Here, ∂xi |x [f] = ∂xi |ϕ(x). Definition. Given f : M → N a C∞ map and x ∈ M we define the push-forward of f at x as follows:

f∗ |x: TxM → Tf(x)N (f∗ |x X)([g]) := X([g ◦ f])

∂ −1 ∂ Remark. With this definition we have ∂xi |x= ϕ∗ ( ∂xi |ϕ(x)).

3 Remark. Let ϕ and ψ be charts about x. Let

ψ ◦ ϕ−1(x1, . . . , xn) = (y1(x1, . . . , xn), . . . , yn(x1, . . . , xn)).

Then ∂ X ∂yk ∂ = (ϕ(x)) . ∂xi x ∂xi ∂yk x k Definition. We define the cotangent to x at M to be the ∗ of TxM. We denote it by Tx M.

i  ∂ Remark. Let {dx |x, 1 ≤ i ≤ n} be the dual basis of ∂xi |x, 1 ≤ i ≤ n , then we ∗ will say that it is the natural basis of Tx M. Remark. Let ϕ and ψ be charts about x. Let

ϕ ◦ ψ−1(y1, . . . , yn) = (x1(y1, . . . , yn), . . . , xn(y1, . . . , yn)).

Then X ∂xi dxi = (ψ(x))dyk . x ∂yk x k Definition. The of M, denoted by TM is defined as follows: a TM := TxM. x∈M

Proposition. Let π : TM → M be defined by π(X) = x if X ∈ TxM. Then there is a unique on TM so that for each coordinate chart (U, ϕ) on M the set U˜ = π−1(U) is open and the map ϕ˜ : U˜ → ϕ(U) × Rn defined by n ! X i ∂ 1 n ϕ˜ v := (ϕ(x), v , . . . , v ) ∂xi i=1 x is a homeomorphism. With this topology, TM is a of dimension 2n and the coordi- nate charts (U,˜ ϕ˜) define a differentiable [C∞,Ck,Cω] structure on TM relative to which π is an open mapping.

Definition. The of M, denoted by T ∗M is defined as follows:

∗ a ∗ T M := Tx M. x∈M

4 ∞ Definition. A C map f : M → N is said to be an inmersion if f∗ is injective at every point.

Definition. An inmersed of a manifold N is a pair (M, f) where f : M → N is an injective inmersion.

Definition. An embedded submanifold of a manifold N is an inmersed sub- manifold (M, f) such that f : M → f(M) is a homeomorphism.

Theorem []. Let M and N be C∞ manifolds of dimension ∞ n. Let f : M → N be a C map. Suppose that xo ∈ M is such that

f∗ |xo : Txo M → Tf(xo)N is an . Then there exists an open neighb. U of xo so that 1. f | U is injective. 2. f(U) is open in N. 3. f −1 : f(U) → U is C∞.

Theorem [ Theorem]. Let M and N be C∞ manifolds with ∞ −1 dimM > dimN and let f : M → N be C . Let yo ∈ f(M) and let Mo = f (yo). Suppose that ∀x ∈ Mo the map f∗ |x is surjective. Then Mo can be endowed with ∞ a C structure relative to which the inclusion map ı : Mo → M is an . Furthermore, dimMo = dimM − dimN. Applications.

1. Sn is a C∞ manifold of dimension n. 2. SL(n, R) = {X ∈ GL(n, R) : det X = 1} is a C∞ manifold of dimension n2 − 1. t ∞ 3. O(n, R) = {X ∈ GL(n, R): X X = Idn} is a C manifold of dimension n(n − 1)/2.

Differential forms

Definition. Let V be a real n-dimensional space and let V ∗ be its dual space. We define the space of alternating k-forms as follows:

k ∗ Λ (V ) = {ω : V ⊕ · · · ⊕ V (k times) → R : ω is linear and alternating}.

k ∗ n!k! Observe that dimΛ (V ) = (n−k)! .

5 Remark. ω is linear and alternating if ω(v1, . . . , vn) is linear in each argument and

π ω(vπ(1), . . . , vπ(k)) = (−1) ω(v1, . . . , vn).

Definition. Let Λ0(V ∗) := R. We define the grassman algebra of V ∗ as follows n M Λ∗(V ∗) := Λk(V ∗). k=0 Observe that with this definition we have dimΛ∗(V ∗) = 2n. Λ∗(V ∗) will be endowed with a graded product operation as follows:

∧ :Λk(V ∗) × Λl(V ∗) → Λk+l(V ∗)(ω, τ) 7→ ω ∧ τ, where 1 X ω ∧ τ(v , . . . , v ) := (−1)πω(v , . . . , v )τ(v , . . . , v ). 1 k+l (k + l)! π(1) π(k) π(k+1) π(k+l) π∈Sk+l

Remark. If A ∈ End(V ∗) we extend A to an endomorphism of Λ∗(V ∗) in the following way:

A˜ :Λ∗(V ∗) → Λ∗(V ∗) A˜(ω ∧ τ) := Aω ∧ Aµ, ω, τ ∈ V ∗ = Λ1(V ∗).

A˜ preserves degrees by construction. Observe that if ω ∈ Λn(V ∗) then A˜ must be a multiple of ω. Actually, Aω˜ = det(A)ω.

Remark. Let M be a manifold and x ∈ M. Now we are able to construct the following spaces:

∗ ∗ Ln k ∗ • Λ (Tx M) := k=0 Λ (Tx M). k ∗ ` k ∗ • Λ (T M) := x∈M Λ (Tx M). ∗ ∗ ` ∗ ∗ • Λ (T M) := x∈M Λ (Tx M). Definition. Choose a chart (U, ϕ) about x with local coordinates (x1, . . . , xn). An k ∗ element ω |x ∈ Λ (Tx M) is called k-form at x and can be written as

X i1 ik ω |x= ai1...ik dx |x ∧ · · · ∧ dx |x .

1≤i1<...

k ∗ n!k! Λ (T M) is a manifold of dimension n + (n−k)! .

6 Definition. We define a k-form on M as a of the bundle π :Λk(T ∗M) → ∞ k ∗ M. That is, a C map ω : M → Λ (T M) so that π ◦ ω = idM . We will denote the space of k-forms on M by Ωk(M).

∗ Ln k 0 ∞ Notation. Ω (M) := k=0 Ω (M) and Ω (M) = C (M, R). Remark. For every k such that k > dimM we have Ωk(M) = 0.

Definition. We define the operator d :Ω0(M) → Ω1(M) by

df |x (X) := X([f]) X ∈ TxM.

With this definition X ∂ i df |x= [f] dx |x . ∂xi i x

Theorem. There exists a unique R- d :Ω∗(M) → Ω∗(M) called the exterior so that 1. d :Ωk(M) → Ωk+1(M). 2. d(f) = df defined as above for f ∈ C∞(M, R). 3. d(ω ∧ τ) = dω ∧ τ + (−1)degωω ∧ τ. 4. d2 = 0.

Definition. Let f : M → N be a C∞ map. We define the pull back of f as the map f ∗ :Ω∗(N) → Ω∗(M) so that

1. f ∗(g) = g ◦ f for f ∈ Ω0(N) = C∞(N, R). ∗ k 2. f ω |x (X1,...,Xk) = ω |f(x) (f∗X1, . . . , f∗Xk) for ω ∈ Ω (N) with k ≥ 1. Properties of the pull-back map.

1. f ∗(ω ∧ τ) = f ∗ω ∧ f ∗τ 2. f ∗(gω + hτ) = f ∗(g) f ∗ω ∧ f ∗(h) f ∗τ 3. (f ◦ g)∗ = g∗ ◦ f ∗

Proposition. Pull-backs and d commute:

d(f ∗ω) = f ∗(dω).

n Proposition. M is orientable iff ∃ ω ∈ Ω (M) such that ω |x6= 0 for all x ∈ M.

7 Definition. An orientation on M is a choice of an [ω] ∈ dim M ∞ + Ω (M) where ω1 ∼ ω2 iff ω1 = fω2 for some f ≥ 0, f ∈ C (M, R ). We denote M endowed with an orientation by [M].

Definition. of n-forms. Let M be an orientable manifold of dimen- sion n.

1. If ω ∈ Ωn(Rn) has compact , and ω = fdx1 ∧ · · · ∧ dxn then Z Z ω := fdx1 . . . dxn. n n R R 2. If ω ∈ Ωn(M) we define Z Z Z X X ∗ −1 ω = ραω := (ϕα) (ραω) [M] α∈a Uα α∈a ϕ(Uα)

where {(Uα, ϕα): α ∈ A} is a positively oriented and {ρα : α ∈ A} is a partition of unity subordinate to {(Uα, ϕα): α ∈ A}.

Definition. A C∞ manifold with boundary is a Hausdorff space with a countable basis of open sets and a differentiable structure {(Uα, ϕα): α ∈ A} such n that it has compatibility on overlaps and ϕα(Uα) is open in H . We will denote the boundary of M by ∂M.

Convention. If Hn is given the orientation [dx1 ∧ · · · ∧ dxn], then ∂Hn is given the orientation [(−1)ndx1 ∧ · · · ∧ dxn].

n−1 Theorem[Stokes]. If ω ∈ Ωc (M) where M is an oriented manifold with oriented boundary ∂M then Z Z dω = ω. [M] [∂M]

Corollary. Let M be a compact manifold without boundary and θ ∈ Ωn(M). Then R if θ is exact we have M θ = 0.

Definition. An open cover {Uα : α ∈ a} of a manifold M is said to be a good n cover if all non-empty intersections Uα0 ∩ · · · ∩ Uαp are diffeomorphic to R . Proposition. Every manifold admits a good cover.

Corollary. Every compact manifold admits a finite good cover.

8

Definition. A is a pair (M, g) where M is a C∞ mani- fold and g is a map that assigns to any x ∈ M a non-degenerate symmetric positive definite bilinear form gx : TxM × TxM → R such that for all X,Y smooth vector fields on M, the map x 7→ gx(X,Y ) is smooth.

1 n ∂ Notation. Let (U, ϕ) be a chart with local coordinates (x , . . . , x ), and let { ∂x1 |x ∂ ,..., ∂xn |x} be the natural basis of TxM. Then we will adopt the following notation:  ∂ ∂  gij(x) := gx , . ∂xi x ∂xj x

ij We will also denote by g the entrances of the inverse of (gij)ij. Proposition. Let M and N be manifolds and let g be a Riemannian on N. Let f : M → N be a C∞ inmersion. Then the map f ∗ defined as below defines a Riemannian metric -f ∗g- on M.

∗ (f g)x(Xx,Yx) := gf(x)(f∗Xx, f∗Yx).

Theorem. Every manifold carries a Riemannian metric.

Examples of Riemannian manifolds.

1. (R, gstand) n ∗ 2. If M ⊂ R then (M, ι gstand) is a Riemannian manifold isometrically immersed in an . Here, ι : M → Rn is the inclusion map. 3. Lie Groups. 4. If (M, g) and (N, h) are Riemannian manifolds, we can endow M × N with the product metric: If π1 : M × N → M and π2 : M × N → M are the projections then the product metric is defined as follows

∗ ∗ g ⊕ h(X,Y ) := π1(g)(X,Y ) + π2(h) (X,Y ).

Definition. Let (M, g) be a Riemannian manifold. An Affine is a map ∇ : Γ(TM) × Γ(TM) → Γ(TM)(X,Y ) 7→ ∇X Y satisfying the following conditions:

9 ∞ 1. ∇fX+gY Z = f∇X Z + g∇Y Z for f, g ∈ C (M, R).

2. ∇X (Y + Z) = ∇X Y + ∇X Z.

3. ∇X (fY ) = f∇X Y + X(f)Y.

Theorem[Fundamental theorem of Riemannian Geometry] . ∇ exists and is uni- que.

i Definition. We define the Γjk : M → R so that

∂ X k ∂ ∂ ∇ j = Γij k . ∂xi ∂x ∂x k Then,   1 X ∂gjl ∂gjk ∂gkl Γi = gil − + . jk 2 ∂xk ∂xl ∂xj l

P i ∂ P i ∂ Remark. Let X = i a ∂xi and Y = i b ∂xi . Then, ! X ∂bj ∂ X ∂ ∇ Y = ai + bjΓk . X ∂xi ∂xj ij ∂xk i k,i

This allows us to define ∇Xx Y |x∈ TxM where Xx ∈ TxM and Y ∈ Γ(TM) as follows: P i ∂ i If Xx = i α ∂xi |x where now α is a number for all i (instead of a function) and Y is as before, then ! ∂bj ∂ ∂ X i X j k ∇Xx Y |x= α + b (x)Γij(x) . ∂xi x ∂xj x ∂xk x i k,i

Definition. Let γ :(a, b) → M. We say that V is a along γ if it is a function that for every t assigns a of the form X ∂ V (t) = vi(t) . ∂xi γ(t) i

Sometimes we will write V |γ(t) instead of V (t).

In particular, we define the vector field γ˙ along the γ as follows  d  γ˙ (t) := γ∗ . dt t

10 Observe that if ϕ ◦ γ = (x1, . . . , xn) we have that

X dxi ∂ γ˙ (t) = (t) . dt ∂xi γ(t) i

Remark. Because of the observations made in previous definition, given X ∈ Γ(TM) and a curve γ it has sense to consider a vector field along γ defined as follows ∇γ˙ X (t) = ∇γ˙ (t)X |γ(t) .

D Theorem. There exists a unique operator dt on vector fields along a given curve γ so that

D D D 1. dt (V + W ) = dt (V ) + dt (W ). D df D ∞ 2. dt (fV ) = dt V + f dt (V ) for f ∈ C (R). D 3. If X ∈ Γ(TM) is such that X |γ(t)= V (t) we have dt (V ) = ∇γ˙ X.

P i ∂ Remark. If V (t) = i v (t) ∂xi |γ(t) then

k i ! DV X X dv X k dx j ∂ (t) = (t) + Γij(γ(t)) (t)v (t) . dt dt dt ∂xk γ(t) k k i,j

Definition. Let M be a differentiable manifold with affine connection ∇. A vector DV field V along a curve γ : I → M is called parallel when dt = 0, for all t ∈ I. Proposition. Let M be a differentiable manifold with an affine connection ∇. Let γ : I → M be a differentiable curve in M and let V0 be a vector tangent to M at

γ(t0), t0 ∈ I (i.e. V0 ∈ Tγ(t0)M). Then there exists a unique parallel vector field V along γ, such that V (t0) = V0. Definition. Let M be a differentiable manifold with an affine connection ∇ and a Riemannian metric h, i. A connection is said to be compatible with the metric h, i, when for any smooth curve γ and any pair of parallel vector fields P and P 0 along γ, we have hP,P 0i = constant. Proposition. Let M be a Riemannian manifold. A connection ∇ on M is compati- ble with a metric if and only if for any vector fields V and W along the differentiable curve γ : I → M we have d DV DW hV,W i = h ,W i + hV, i, t ∈ I. dt dt dt

11 Corollary. A connection ∇ on a Riemannian manifold M is compatible with the metric if and only if

X(hY,Zi) = h∇X Y,Zi + hY, ∇X Zi,X,Y,Z ∈ Γ(TM).

Definition. An affine connection ∇ on a smooth manifold M is said to be symme- tric when ∇X Y − ∇Y X = [X,Y ] for all X,Y ∈ Γ(TM).

Theorem, (Levi-Civita). Given a Riemannian manifold M, there exists a unique affine connection ∇ on M satisfying the conditions:

1. ∇ is symmetric. 2. ∇ is compatible with the Riemannian metric.

n k ∂ Remark. For the Euclidean space R we have Γij = 0. As a consequence ∇ ∂ ∂x = ∂xi j n Dv P dvk ∂ P ∂ 0. Thus, in , = and ∇X Y = X(yk) . R dt k dt ∂xk k ∂xk

Dγ˙ Definition. γ :(a, b) → R is a if dt = 0 . P dxi ∂ 1 n Observe that since γ˙ (t) = i dt (t) ∂xi |γ(t) (where ϕ ◦ γ = (x , . . . , x )) then

2 k i j ! Dγ˙ X X d x X k dx dx ∂ (t) = (t) + Γij(γ(t)) (t) (t) . dt dt2 dt dt ∂xk γ(t) k k i,j

dγ Remark. If γ :(a, b) → R is a geodesic, | dt | is constant. Lemma. there exists a unique vector field G on TM whose trajectories are of the form t → (γ(t), γ0(t)), where γ is a geodesic on M.

Definition. The vector field G defined above is called the geodesic field on TM and its flow is called the geodesic on TM. Lemma. If the geodesic γ(t, q, v) is defined on the (−δ, δ), then the geodesic γ(t, q, av), a ∈ R, a > 0, is defined on the interval (−δ/a, δ/a) and γ(t, q, av) = γ(at, q, v).

Proposition. Given p ∈ M, there exist a neighborhood V of p in M, a number ε > 0 and a C∞ mapping γ :(−2, 2) × U → M,

U = {(q, w) ∈ TM; q ∈ V, w ∈ TqM, |w| < ε} ,

12 such that t → γ(t, q, w), t ∈ (−2, 2), is the unique geodesic of M which, at the instant t = 0, passes through q with w, for every q ∈ V and for every w ∈ TqM, with |w| < ε.

Definition. We define the exponential map exp : U → M by  v  exp(q, v) = γ(1, q, v) = γ |v|, q, , (q, v) ∈ U. |v|

In most applications we shall utilize the restriction of exp to an open subset of the tangent space TqM, that is,

expq : Bε(0) ⊂ TqM → M,

expq(v) = exp(q, v).

Proposition. Given q ∈ M, there exists an ε > 0 such that expq : Bε(0) ⊂ TqM → M is a diffeomorphism of Bε(0) onto an open subset of M.

Definition. A segment of the geodesic γ :[a, b] → M is called minimizing if `(γ)) ≤ `(c), where c is an arbitrary piecewise differentiable curve joining γ(a) to γ(b).

Definition. A parametrized in M is a differentiable mapping s : A ⊂ R2 → M.A vector field along s is a mapping which associates to each q ∈ A a vector V (q) ∈ Ts(q)M, and which is differentiable in the following sense: if f is a differentiale function on M, then the mapping q → V (q)f is differentiable. If V is a vector field along s : A → M, let us define the covariant DV DV DV ∂u and ∂v in the following way. ∂u (u, v0) is the along the curve u → s(u, v0) of the restriction of V to this curve. Lemma(). If M is a differentiable manifold with a symmetric connec- tion ans s : A → M is a parametrized surface then: D ∂s D ∂s = . ∂v ∂u ∂u ∂v

Lemma(Gauss). Let p ∈ M and let v ∈ TpM such that expq v is defined. Let w ∈ TpM ∼ Tv(TpM). Then

hdv expp(v), dv expp(w)i = hv, wi.

13 Definition. If expp is a diffeomorphism of a neighborhood V of the origin in TpM, expp V = U is called a neighborhood of p. If Bε(0) is such that Bε(0) ⊂ V , we call expp B(0) = Bε(p) the normal , or geodesic ball. Proposition. Let p ∈ M, U a normal neighborhood of p, and B ⊂ U a normal ball of center p. Let γ : [0, 1] → B be a geodesic segment with γ(0) = p. If c : [0, 1] → M is any piecewise differentiable curve joining γ(0) to γ(1) then `(γ) ≤ `(c) and if equality holds then γ([0, 1]) = c([0, 1]).

Theorem. For any p ∈ M there exist a neighborhood W of p and a number

δ > 0, such that, for every q ∈ W , expq is a diffeomorphism on Bδ(0) ⊂ TqM and W ⊂ expq(Bδ(0)), that is, W is a normal neighborhood of each of its points. Definition. The R of a Riemannian manifold M is a correspondence that associates to every pair X,Y ∈ Γ(TM) a mapping

R(X,Y ) : Γ(TM) → Γ(TM),

R(X,Y )Z = ∇Y ∇X Z − ∇X ∇Y Z + ∇[X,Y ]Z,Z ∈ Γ(TM), where ∇ is the Riemannian connection of M. h i Remark. In particular, since ∂ , ∂ = 0, we obtain ∂xi ∂xj

 ∂ ∂  ∂   ∂ R , = ∇ ∂ ∇ ∂ − ∇ ∂ ∇ ∂ . ∂xi ∂xj ∂xk ∂xj ∂xi ∂xi ∂xj ∂xk That is, the curvature measures the non-commutativity of the covariant derivative.

Proposition. The curvature R of a Riemannian manifold has the following pro- perties:

1. R is bilinear in Γ(TM) × Γ(TM). 2. For any X,Y ∈ Γ(TM), the curvature operator R(X,Y ) : Γ(TM) → Γ(TM) is linear.

Proposition (Bianchi Identity).

R(X,Y )Z + R(Y,Z)X + R(Z,X)Y = 0.

14 Proposition. Let σ ⊂ TpM be a two-dimensional subspace of the tangent space TpM and let x, y ∈ σ be two linearly independent vectors. Then

(x, y, x, y) K(x, y) = |x ∧ y|2 does not depend on the choice of the vectores x, y ∈ σ.

Definition. Given a point p ∈ M and a two-dimensional subspace σ ⊂ TpM, the K(x, y) = K(σ), where {x, y} is any basis of σ, is called the of σ at p.

Definition. Let x = zn be a unit vector in TpM; we take an orthonormal basis {z1, . . . , zn−1} of the hyperplane in TpM orthogonal to x and consider the following averages:

1. : 1 X Ric (x) = hR(x, z )x, z i, i = 1, . . . , n − 1. p n − 1 i i i

2. : 1 X 1 X R(p) = Ric (z ) = hR(z , z )z , z i, j = 1, . . . , n − 1. n p j n(n − 1) i j i j j ij

Lemma. D D D D ∂f ∂f  V − V = R , V. ∂t ∂s ∂s ∂t ∂s ∂t

Definition. Let γ : [0, a] → M be a geodesic in M. A vector field J along γ is said to be a Jacobi Field if it satisfies the Jacobi equation D2J (t) + R(γ0(t),J(t))γ0(t) = 0, for allt ∈ [0, a]. dt2

Example. Let M be a Riemannian manifold of constant sectional curvature K, and let γ : [0, `] → M be a normalized geodesic on M. Then the Jacobi equation ca be written as D2J + KJ = 0. dt2

15 Let w(t) be a parallel field along γ with hγ0(t), w(t)i = 0 and |w(t)| = 1. Then √  sin(t K)  √ w(t), ifK > 0,  K J(t) = tw(t), ifK = 0, √  sinh(t K)  √ w(t), ifK < 0,  K is a solution of the Jacobi equation with initial conditions J(0) = 0, J 0(0) = w(0).

Theorem. Let γ : [0, a] → M be a geodesic. Then a Jacobi field J along γ with J(0) = 0 is given by

0 J(t) = Dtγ0(0) expp(tJ (0)), t ∈ [0, a].

Definition. Let γ : [0, a] → M be a geodesic. the point γ(t0) is said to be con- jugate to γ(0) along γ, if there exists a Jacobi field J along γ, not identically zero, with J(0) = 0 = J(t0). The maximum number of such linearly independent fields is called the multiplicity of the conjugate point γ(t0). The set of (first) conjugate points to the point p ∈ M, for all the that start at p, is called the conjugate locus of p and is denoted by C(p). Proposition. Let γ : [0, a] → M be a geodesic and put γ(0) = p. The point 0 q = γ(t0), t0 ∈ (0, a], is conjugate to p along γ if and only if v0 = t0γ (0) is a critical point of expp. In , the multiplicity of q as a conjugate of p is equal to the dimension of the kernel of the linear map Dv0 expp. Corollary. Suppose that J(0) = 0. Then hJ 0(0), γ0(0)i = 0 if and only if hJ, γ0i(t) = 0; in particular, the space of Jacobi fields J with J(0) = 0 and hJ, γ0i = 0 has dimension equal to n − 1.

Definition. A Riemannian manifold is said to be extendible if there exists a riemannian manifold M 0 such that M is isometric to a proper open subset of M 0.

Definition. A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map expp is defined for all v ∈ TpM, i.e., if any geodesic γ(t) starting from p is defined for all values of t ∈ R. Proposition. If M is geodesically complete then M is non-extendible.

Definition. The distance d(p, q) is defined by d(p, q) = infimum of the lengths of all fp,q where fp,q is a piecewise differentiable curve joining p to q. Proposition. With the distance d, M is a metric space.

16 Proposition. The topology induced by d on M coincides with the original topology on M.

Corollary. If p0 ∈ M, the function f : M → R given by f(p) = d(p, p0) is continuous.

Theorem (Hopf-Rinow). Let M be a Riemannian manifold and let p ∈ M. the following assertions are equivalent:

1. expp is defined on all of TpM. 2. The closed and bounded sets of M are compact. 3. M is complete as a metric space. 4. M is geodesically complete.

5. There exists a sequence of compact subsets Kn ⊂ M, Kn ⊂ Kn+1 and ∪nKn = M, such that if qn ∈/ Kn then d(p, qn) → ∞. In addition, any of the statements above implies that for any q ∈ M there exists a geodesic γ joining p to q with `(γ) = d(p, q).

Corollary. If M is compact then M is complete.

Corollary. A closed submanifold of a complete Riemannian manifold is complete in the induced metric; in particular, the closed of Euclidean space are complete.

Lemma. Let M be a complete Riemannian manifold with K(p, σ) ≤ 0, for all p ∈ M and for all σ ⊂ TpM. Then for all p ∈ M, the conjugate locus C(p) =; in particular the exponential map expp : TpM → M is a local diffeomorphism. Lemma. Let M be a complete Riemannian manifold and let f : M → N be a local diffeomorphism onto a Riemannian manifold N which has the following property: for all p ∈ M and for all v ∈ TpM, we have |Dpf(v)| ≥ |v|. Then f is a covering map.

Theorem (Hadamard). Let M be a complete Riemannian manifold with K(p, σ) ≤ n 0, for all p ∈ M and for all σ ⊂ TpM. Then M is diffeomorphic to R , n = dimM; more precisely expp : TpM → M is a diffeomorphism.

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