# 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.
• 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, then $d\omega=\sum_{i_1

• 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 $f\in\mathbf{F}(M)$.