Math 327 Final Summary 20 February 2015 1. Manifold A subset M ⊂ Rn is Cr a manifold of dimension k if for any point p 2 M, there is an open set U ⊂ Rk and a Cr function φ : U ! Rn, such that (1) φ is injective and φ(U) ⊂ M is an open subset of M containing p; (2) φ−1 : φ(U) ! U is continuous; (3) D φ(x) has rank k for every x 2 U. Such a pair (U; φ) is called a chart of M. An atlas for M is a collection of charts f(Ui; φi)gi2I such that [ φi(Ui) = M i2I −1 (that is the images of φi cover M). (In some texts, like Spivak, the pair (φ(U); φ ) is called a chart or a coordinate system.) We shall mostly study C1 manifolds since they avoid any difficulty arising from the degree of differen- tiability. We shall also use the word smooth to mean C1. If we have an explicit atlas for a manifold then it becomes quite easy to deal with the manifold. Otherwise one can use the following theorem to find examples of manifolds. Theorem 1 (Regular value theorem). Let U ⊂ Rn be open and f : U ! Rk be a Cr function. Let a 2 Rk and −1 Ma = fx 2 U j f(x) = ag = f (a): The value a is called a regular value of f if f is a submersion on Ma; that is for any x 2 Ma, D f(x) has r rank k. Then Ma is a C manifold of dimension n − k. The set Ma is called a level set of f; thus any level set of a regular value of f is a manifold. The proof of this theorem easily follows from the implicit function theorem. See PS5 for some examples of manifolds. If (U; φ) and (V; ) are two charts of M and φ(U) \ (V ) 6= ; then the function −1 ◦ φ : φ−1( (V )) ! V is a diffeomorphism. It is called a transition function between the two charts. 2. Manifold with Boundary k k k k Let H = f(x1; : : : ; xk) 2 R j xk ≥ 0g be the upper half space in R . Note that the boundary of H as k k k k−1 a subset of R is the set Bd H = f(x1; : : : ; xk) 2 R j xk = 0g = R × f0g. A subset M ⊂ Rn is a Cr manifold with boundary of dimension k if for any point p 2 M, there is an 1 open set U ⊂ Hk or U ⊂ Rk and a Cr function φ : U ! Rn, such that (1) φ is injective and φ(U) ⊂ M is an open subset of M containing p; (2) φ−1 : φ(U) ! U is continuous; (3) D φ(x) has rank k for every x 2 U. 1 m p r r Recall that if S ⊂ R is an arbitrary subset then we say f : S ! R is C if there is an open set V ⊃ S and a C p function g : V ! R such that f(x) = g(x) for all x 2 S. 1 2 Again the pair (U; φ) is called a chart for M, and a collection of charts which cover M is called an atlas. Note that a chart (U; φ) for a manifold with boundary is defined in the exact same way as a chart for a manifold (without boundary), however now the open set can come from Hk. Let M ⊂ Rn be a k-dimensional manifold with boundary. Then p 2 M is an interior point if there is a chart (U; φ) of M with U ⊂ Rk open and p 2 φ(U). Any point of M which is not an interior point is called a boundary point and the set of all boundary points of M denoted by @M. In fact @M ⊂ Rn is a manifold (without boundary) of dimension k − 1 (one less than dimension of M). A point p 2 M is in @M if there is a chart (U; φ) where U ⊂ Hk is open and p = φ(x) for some k x 2 U \ Bd H (i.e. if x = (x1; : : : ; xk) then xk = 0). Notice that the concept of boundary of a manifold M in Rn denoted @M is different from the concept of the boundary of a subset S of Rn denoted Bd S. Similarly the interior of M namely M − @M is a different concept to the interior of a set S denoted Int S. One can generalise Theorem 1 as follows: Theorem 2. Let U ⊂ Rn be open and f : U ! Rk be a Cr submersion, then M = fx 2 U j f1(x) = ::: = fk−1(x) = 0; fk(x) ≥ 0g is a Cr manifold with boundary of dimension n − k + 1 and @M = fx 2 U j f(x) = 0g: 3. Maps between Manifolds Let M ⊂ Rm and N ⊂ Rn be C1 manifolds, and f : M ! N a function. f is differentiable if for any two charts (U; φ) of M and (V; ) of N, the composite function −1 ◦ f ◦ φ is differentiable. f is smooth if all the composite functions are smooth. f : M ! N is a diffeomorphism if f is smooth and there is a smooth function g : N ! M for which f ◦ g = IdN and g ◦ f = IdM . Proposition 3. f : M ! N is a diffeomorphism if and only if f is a smooth bijection and for any for any two charts (U; φ) of M and (V; ) of N, the derivative of the composite function −1 ◦ f ◦ φ is invertible at every point. This proposition follows easily from the inverse function theorem which will say that the inverse of f is also smooth. For manifolds with boundary the definitions are completely analogous. It is easy to see that in that case a diffeomorphism takes boundary of domain manifold to the boundary of the range manifold and induces a diffeomorphism of the two boundaries. See assignment 10 for examples of diffeomorphic manifolds. 4. Tangent spaces and Vector fields Let p 2 Rn, then the tangent space of Rn at p is the set fpg × Rn. n n TpR = f(p; v) j v 2 R g 3 Some times the vector (p; v) is also written as vp and usually the vector v is written as a column vector whereas the point p is a point of Rn. The tangent space forms a vector space. If f : Rn ! Rn is a differentiable map then the derivative of f gives a map between tangent spaces. If n n m p 2 R , then define f∗ : TpR ! Tf(p)R as f∗((p; v)) = (f(p); D f(p)v): Note that @vf(p) = D f(p)v is the directional derivative of f at p along v. Hence the tangent vectors are directions along which we differentiate functions. n m m k From the chain rule it follows that if g : R ! R and f : R ! R then (f ◦ g)∗ = f∗ ◦ g∗. Let M ⊂ Rn be a manifold of dimension k, and p 2 M. Suppose (U; φ) is a chart of M such that p = φ(x) then the tangent space of M at p is defined as k TpM = φ∗(TxR ): n 2 Since Dφ(x) has rank k, so TpM is a vector subspace of TpR of dimension k. Theorem 4. 3 If f : Rn ! Rk and a 2 Rk is a regular value of f, then M = f −1(a) is a manifold. Let p 2 Ma, then n k TpM = ker(f∗ : TpR ! TaR ): A differentiable function f : M ! N between manifolds induces a linear map between tangent spaces. If p 2 M then we get an induced linear map f∗ : TpM ! Tf(p)N which is the linearisation of f at p. The set n [ n n n (1) T R = TpR = R × R ; n p2R is called the tangent bundle of Rn and is naturallyidentified with R2n. The set [ n TM = TpM ⊂ T R p2M is called the tangent bundle of M. Although we haven't proved this, but it is a manifold of dimension 2k in Rn × Rn. If M is a manifold, a continuous (respectively smooth) vector field X on M is a continuous (respectively smooth) function X : M ! TM, such that X(p) 2 TpM for each p 2 M. Thus a vector field is a function form the manifold to its tangent bundle which assigns a tangent vector at each point on M. If U ⊂ Rn is an open subset (hence an n dimensional manifold) then we have the standard vector fields n on U. Let e1; : : : ; en be the standard basis of R . Then define the vector fields E1;:::;En on U by E1(p) = (p; e1);:::;En(p) = (p; en): These vector fields form a frame for the tangent bundle of U in the sense that at each point the vector fields form a basis for the tangent space at that point. Any other vector field X can be written as X(p) = f1(p)E1(p) + ··· + fn(p)En(p): n Thus a vector field on U is equivalent to a function f = (f1; : : : ; fn): U ! R . The vector field is continuous or smooth if and only if the corresponding function is continuous or smooth. 2In ordinary vector calculus the tangent space to a point on a manifold is an affine space. For example the tangent line to a point on a curve or the tangent plane to a point on a surface.