K & N Definitions Section 1.1

Notation
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Chapter 1

1. Differentiable Manifolds

  • Let S be a topological space. A pseudogroup of transformations on S is a set \Gamma of tansformations satisfying the following axioms:
    1. Each f\in\Gamma is a homeomorphism of an open subset \text{dom}(f) of S onto the open subset \text{range}(f) of S.
    2. If f\in\Gamma, then the restriction of f to an arbitrary open subset of the domain of f is in \Gamma.
    3. Let U=\cup_i U_i where each U_i is an open set of S. A homeomorphism f of U onto an open set of S belongs to \Gamma if the restriction of f to U_i is in \Gamma for every i.
    4. For every open set U of S, the identity transformation of U is in \Gamma.
    5. If f\in\Gamma, then f^{-1}\in\Gamma.
    6. If f\in\Gamma is a homeomorphism of U onto V and f'\in\Gamma is a homeomorphism of U' onto V' and if V\cap U' is non-empty, then the homeomorphism f'\circ f of f^{-1}(V\cap U') onto f'(V\cap U') is in \Gamma.

  • A mapping f of an open set of \mathbb{R}^n into \mathbb{R}^m is said to be of class C^r, r=1,2,\cdots,\infty, if f is continuously r times differentiable.
    • The pseudogroup of transformations of class C^r, denoted \Gamma^r(\mathbb{R}^n), is the set of homeomorphisms f of an open set of \mathbb{R}^n onto an open set of \mathbb{R}^n such that both f and f^{-1} are of class C^r.
    • The colleciton of those f\in\Gamma^r(\mathbb{R}^n) whose Jacobians are positive everywhere is a subpseudogroup of \Gamma^r(\mathbb{R}^n) called the pseudogroup of orientation-preserving transformations of class C^r of \mathbb{R}^n and is denoted \Gamma^r_0(\mathbb{R}^n).

  • An atlas of a topological space M compatible with a pseudogroup \Gamma is a family of pairs (U_i,\varphi_i) called charts which satisfy the following:
    1. Each U_i is an open set of M and \cup_i U_i=M
    2. Each \varphi_i is a homeomorphism of U_i onto an open set of S.
    3. Whenever U_i\cap U_j is nonempty, the mapping \varphi_j\circ\varphi_i^{-1} of \varphi_i(U_i\cap U_j) onto \varphi_j(U_i\cap U_j) is an element of \Gamma.

  • A complete atlas of M compatible with \Gamma is an atlas of M compatible with \Gamma which is not contained in any other atlas of M compatible with \Gamma.

  • A differentiable manifold of class C^r is a Hausdorff space with a fixed complete atlas compatible with \Gamma^r(\mathbb{R}^n). An oriented differentiable manifold of class C^r is a Hausdorff space with a fixed complete atlas compatible with \Gamma^r_0(\mathbb{R}^n).

  • An orientable differentiable structure is a differentiable structure of class C^r obtained from an oriented differentiable structure of class C^r.

  • An allowable chart of a structure is a chart which belongs to the fixed complete atlas defining the structure.

  • Given an allowable chart (U_i,\varphi_i) of an n-dimensional manifold M of class C^r, the system of functions x_1\circ\varphi_i,\cdots,x_n\circ\varphi_i defined on U_i is called a local coordinate system in U_i and U_i is called a coordinate neighborhood.

  • Given two manifolds M and M' of class C^r, a mapping f:M\to M' is said to be differentiable of class C^k, k\leq r, if, for every chart (U_i,\varphi_i) of M and every chart (V_j,\psi_j) of M' such that f(U_i)\subset V_i, the mapping \psi_j\circ f\circ \varphi_i^{-1} of \varphi_i(U_i) into \psi_j(V_j) is differentiable of class k.
    • A maping of class C^\infty is referred to as a (differentiable) mapping.

  • A differentiable curve of class C^k in M is a differentiable mapping of class C^k of a closed interval [a,b] of \mathbb{R} into M, that is, a restriction of a differentiable mapping of class C^k of an open interval containing [a,b] into M.

  • Given a manifold M and a point p\in M, let \mathbf{F}(p) be the algebra of differentiable functions of class C^1 defined in a neighborhood of p.

  • Let x(t) be a curve of class C^1, a\leq t\leq b, such that x(t_0)=p. The vector tangent to the curve x(t) at p is a mapping X:\mathbf{F}(p)\to \mathbb{R} defined by Xf=(df(x(t))/dt)_{t_0}, i.e. Xf is the derivative of f in the direction of the curve x(t) at t=t_0.

  • The set of tangent vectors at p is called the tangent space of M at p and is denoted by T_p(M) or T_p.
    • Given a tangent vector \sum_j \xi_j(\partial/\partial u_j)_p, the n-tuple \xi_1,\ldots,\xi_n is called the components of the vector with respect to the local coordinate system u_1,\ldots,u_n.

  • A vector field X on a manifold M is an assignment of a vector X_p to each point p on M. In particular, if f\in\mathbf{F}(p) is a differentiable function on M, then Xf is a function on M defined by (Xf)(p)=X_p f.
    • A vector field is called differentiable if Xf is differentiable for every differentiable function f.
    • Givne a local coordinate system u_1,\ldots,u_n, a vector field X may be expressed as X=\sum_j \xi_j(\partial/\partial u_j), where \xi_j are again called components of X with respect to u_1,\ldots,u_n.

  • Let \mathbf{X}(M) be the set of all differentiable vector fields on M.

  • Given X,Y\in\mathbf{X}(M), define the bracket [X,Y]:R_M\to R_M of X,Y as a mapping from the ring of functions R_M on M to itself by [X,Y]f=X(Yf)-Y(Xf).

  • For p\in M, the dual vector space T_p^*(M) of T_p(M) is called the space of covectors at p.
    • An assignment of a covector at each point p is called a 1-form or a differential form of degree 1.
    • The functions (defined in the neighborhood of p) f_j are called the components of the 1-form \omega=\sum_j f_j\,du_j (defined in a neighborhood of p) with respect to u_1,\ldots,u_n.
    • The 1-form \omega is called differentiable if each f_j is differentiable.

  • For each function f on M, the total differential (df)_p of f at p is defined by \langle(df)_p,X\rangle=Xf for X\in T_p(M) where \langle\,\,,\,\rangle denotes the value of the first entry on the second entry as a linear functional on T_p(M).

  • Recall that an exterior algebra \Lambda(V) over a vector space V is, roughly, an algebra whose product is the exterior product \wedge. Let \Lambda T_p^*(M) be the exterior algebra over T_p^*(M). An r-form \omega is an assignment of an element of degree r in \Lambda T_p^*(M) to each point p of M.
    • In terms of a local coordinate system u_1,\ldots,u_n, \omega can be expressed uniquely as \omega=\sum_{i_1<i_2<\cdots<i_r} f_{i_1\cdots i_r} du_{i_1}\wedge\cdots\wedge du_{i_r}.
    • The r-form \omega is called differentiable if the components f_{i_1\cdots i_r} are all differentiable.

  • Denote by \mathbf{D}^r=\mathbf{D}^r(M) the collection of all differentiable r-forms on M for each r=0,1,\ldots,n. Let \mathbf{D}=\sum_r \mathbf{D}^r(M).

  • Exterior Differentiation d on \mathbf{D} can be characterized as follows:
    1. d is an \mathbb{R}-linear mapping of \mathbf{D}(M) into itself such that d(\mathbf{D}^r)\subset\mathbf{D}^{r+1}.
    2. For a function f\in\mathbf{D}^0, df is the total differential.
    3. If \omega\in\mathbf{D}^r and \pi\in\mathbf{D}^s, then d(\omega\wedge\pi)=d\omega\wedge\pi + (-1)^{r}\omega\wedge d\pi.

    In terms of a local coordinate system, if \omega=\sum_{i_1<i_2<\cdots <i_r}f_{i_1\cdots i_r}du_{i_1}\wedge\cdots du_{i_r}, then d\omega=\sum_{i_1<i_2<\cdots <i_r}df_{i_1\cdots i_r}\wedge du_{i_1}\wedge\cdots du_{i_r}

  • Given a mapping f of a manifold M into another manifold M', the differential at p\in M of f is the linear mapping f_* of T_p(M) into T_{f(p)}(M') defined as follows: For each X\in T_p(M), choose a curve x(t) in M such that X is the vector tangent to x(t) at p=x(t_0). Then f_*(X) is the vector tangent to the curve f(x(t)) at f(p)=f(x(t_0)). When it is necessary to specify the point p, the notation (f_*)_p is used.
    • The transpose of (f_*)_p is a linear mapping of T_{f(p)}^*(M') into T_p^*(M).

  • For any r-form \omega' on M', define the r-form f^*\omega' on M by (f^*\omega')(X_1,\ldots, X_r)=\omega'(f_*X_1,\ldots,f_*X_r) for X_1,\ldots,X_r\in T_p(M).
    • Note that exterior differentiation commutes with f^* so that d(f^*\omega')=f^*(d\omega').

  • A mapping f of M into M' is said to be of rank r at p\in M if the dimension of f_*(T_p(M)) is r.
    • If the rank of f at p is equal to n=\dim(M), (f_*)_p is injective and \dim M\leq \dim M'.
    • If the rank of f at p is equal to n'=\dim(M'), (f_*)_p is surjective and \dim M\geq \dim M'.

  • A mapping f of M into M' is called an immersion if (f_*)_p is injective for every point p in M. Here, we say that M is immersed in M' by f or that M is an immersed submanifold of M'.
    • When an immersion f is inejctive, it is called an imbedding of M into M' and it is said that M is a(n) (imbedded) submanifold of M'.

  • An open subset M of a manifold M' considered as a submanifold of M' in a natural manner is called an open submanifold of M'.

  • A diffeomorphism of a manifold M onto another manifold M' is a homeomorphism \varphi such that both \varphi and \varphi^{-1} are differentiable.
    • A diffeomorphism of M onto itself is called a (differentiable) transformation of M.

  • A transformation \varphi of M induces an automorphism \varphi^* of the algebra \mathbf{D}(M) of differential forms on M and, in particular, an automorphism of the algebra \mathbf{F}(M) of functions on M: (\varphi^* f)(p)=f(\varphi(p)) for f\in\mathbf{F}(M), p\in M.
    • It also induces an automorphism \varphi_* of the Lie algebra \mathbf{X}(M) of vector fields by (\varphi_* X)_p=(\varphi_*)_q(X_q), where \varphi(q)=p, X\in\mathbf{X}(M).
    • The relation between \varphi_* and \varphi^* is given by the following: \varphi^*((\varphi_* X)f)=X(\varphi^* f) for X\in\mathbf{X}(M) and f\in\mathbf{F}(M).

One thought on “K & N Definitions Section 1.1

  1. Pingback: Stir Crazy | riemannian hunger

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