Differential Geometry Differentiable Manifolds Definition of topological manifold : It is a topological space (E; τ) so that 1. It is Hausdorff. 2. 8x 2 E there exists (U; ') with U open and x 2 U, such that ' : U ! '(U) is a homeomorphism. The pair (U; ') is called chart and the real numbers (x1; : : : ; xn) = '(x) are called local coordinates. 3. (E; τ) has a countable basis of open sets. Remark. Topological Manifolds are paracompact, i.e, every open cover has a locally finite refinement. Remark. Paracompactness implies having Partition of unity. Definition. A differentiable [C1;Ck;C!] structure on a topological ma- nifold M is a family of charts U = f(Uα;'αg so that S 1. Uα = M −1 2. If Uα \ Uβ 6= ; then 'β ◦ 'α : 'α(Uα \ Uβ) ! 'β(Uα \ Uβ) are differentiable [C1;Ck;C!] with differentiable [C1;Ck;C!] inverse. In this case we say that (Uα;'α) and (Uβ;'β) are compatible. 3. Completness property: If (V; ) is a chart which is compatible with every (Uα;'α) 2 U then (V; ) 2 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 f(Vα; α): α 2 Ag be a covering of M by C1-compatible coordinate charts. Then 9! C1 structure containing the charts f(Vα; α): α 2 Ag: Examples of topological manifolds: 1. (Rn; Id) 2. If M is a C1 manifold and N ⊂ M is an open subset then N is a C1 manifold too. 1 3. By the previous example GL(n; R) is a manifold. 4. If M and N are C1 manifolds so is M × N. 5. S2 is a C! manifold. X Definition of open equivalence relation: Let π : X ! ∼ be the projection S −1 and [U] := x2U fπ (x)g. 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 = f(x; y): x ∼ yg. 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 9t 6= 0 so that x = ty. n−1 R −0 Pn(R) := ∼ .) 2. G(n; k) is a C1 manifold of dimension k(n−k). ("G(n; k) = fk-dim’l subspaces n F (n;k) kn of R g", 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 [C1;Ck;C!] map if 8x 2 W 9(U; ') coordinate chart (with x 2 U) so that f ◦ '−1 : '(W \ U) ! R is C1. Definition. Let A ⊂ M. We say that f : A ! R is C1 if it has a C1 extension to an open set U ⊂ M such that A ⊂ U. Definition. Let M and N be differentiable [C1;Ck;C!] manifolds. We will say that f : M ! N is a differentiable [C1;Ck;C!] map if 8x 2 M 9(U; ') coordinate chart with x 2 U and (V; ) coordinate with f(x) 2 V so that f(U) ⊂ V and ◦f ◦'−1 : '(W \U) ! R is differentiable [C1;Ck;C!] as a map of euclidean spaces. 2 Definition. Let M and N be C1 manifolds and f : M ! N be a C1 map. We define the rank of f as the rank of ◦ f ◦ '−1 where (U; ') and (V; ) are charts as h @yi i −1 1 n above. That is, (rank f)x := rank @xj j'(x) where ◦ f ◦ ' = (y ; : : : ; y ). Definition. Let f; g be C1 in a neighb. of x 2 M. We will say that f ∼ g if 9U open so that f(y) = g(y) 8y 2 U. The class [f] is called germ of C1 function at x. The set of germs at x will be denoted by C1(x; R). Definition. Let x 2 M and X : C1(x; R) ! R. If for every chart (U; ') about x we have that there exist a1; : : : ; an 2 R so that n X @ −1 X([f]) = ai (f ◦ ' ) ; @x '(x) i=1 i we will say that X is a tangent vector at x . Observation. If the equation holds for some chart (U; ') about x, then it holds for every C1-compatible chart overlapping at x. Properties of tangent vectors: 1. X([f] + [g]) = X([f]) + X([g]) 2. X([λf]) = λX([f]) 8λ 2 R 3. X([f][g]) = X([f])[g] + [f]X([g]) Definition. The tangent space to M at x is the vector space 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; ') about x 2 M the basis @xi x ; 1 ≤ i ≤ n is called the natural basis of TxM associated to this chart. @ @f◦'−1 Here, @xi jx [f] = @xi j'(x). Definition. Given f : M ! N a C1 map and x 2 M we define the push-forward of f at x as follows: f∗ jx: TxM ! Tf(x)N (f∗ jx X)([g]) := X([g ◦ f]) @ −1 @ Remark. With this definition we have @xi jx= '∗ ( @xi j'(x)). 3 Remark. Let ' and be charts about x. Let ◦ '−1(x1; : : : ; xn) = (y1(x1; : : : ; xn); : : : ; yn(x1; : : : ; xn)): Then @ X @yk @ = ('(x)) : @xi x @xi @yk x k Definition. We define the cotangent bundle to x at M to be the dual space ∗ of TxM. We denote it by Tx M. i @ Remark. Let fdx jx; 1 ≤ i ≤ ng be the dual basis of @xi jx; 1 ≤ i ≤ n , then we ∗ will say that it is the natural basis of Tx M. Remark. Let ' and be charts about x. Let ' ◦ −1(y1; : : : ; yn) = (x1(y1; : : : ; yn); : : : ; xn(y1; : : : ; yn)): Then X @xi dxi = ( (x))dyk : x @yk x k Definition. The tangent bundle of M, denoted by TM is defined as follows: a TM := TxM: x2M Proposition. Let π : TM ! M be defined by π(X) = x if X 2 TxM. Then there is a unique topology on TM so that for each coordinate chart (U; ') on M the set U~ = π−1(U) is open and the map '~ : U~ ! '(U) × Rn defined by n ! X i @ 1 n '~ v := ('(x); v ; : : : ; v ) @xi i=1 x is a homeomorphism. With this topology, TM is a topological manifold of dimension 2n and the coordi- nate charts (U;~ '~) define a differentiable [C1;Ck;C!] structure on TM relative to which π is an open mapping. Definition. The cotangent bundle of M, denoted by T ∗M is defined as follows: ∗ a ∗ T M := Tx M: x2M 4 1 Definition. A C map f : M ! N is said to be an inmersion if f∗ is injective at every point. Definition. An inmersed submanifold 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 [Inverse function Theorem]. Let M and N be C1 manifolds of dimension 1 n. Let f : M ! N be a C map. Suppose that xo 2 M is such that f∗ jxo : Txo M ! Tf(xo)N is an isomorphism. Then there exists an open neighb. U of xo so that 1. f j U is injective. 2. f(U) is open in N. 3. f −1 : f(U) ! U is C1. Theorem [Implicit function Theorem]. Let M and N be C1 manifolds with 1 −1 dimM > dimN and let f : M ! N be C . Let yo 2 f(M) and let Mo = f (yo). Suppose that 8x 2 Mo the map f∗ jx is surjective. Then Mo can be endowed with 1 a C structure relative to which the inclusion map { : Mo ! M is an embedding. Furthermore, dimMo = dimM − dimN. Applications. 1. Sn is a C1 manifold of dimension n. 2. SL(n; R) = fX 2 GL(n; R) : det X = 1g is a C1 manifold of dimension n2 − 1. t 1 3. O(n; R) = fX 2 GL(n; R): X X = Idng 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 ) = f! : V ⊕ · · · ⊕ V (k times) ! R : ! is linear and alternatingg: k ∗ n!k! Observe that dimΛ (V ) = (n−k)! . 5 Remark. ! is linear and alternating if !(v1; : : : ; vn) is linear in each argument and π !(vπ(1); : : : ; vπ(k)) = (−1) !(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 ∗)(!; τ) 7! ! ^ τ; where 1 X ! ^ τ(v ; : : : ; v ) := (−1)π!(v ; : : : ; v )τ(v ; : : : ; v ): 1 k+l (k + l)! π(1) π(k) π(k+1) π(k+l) π2Sk+l Remark.