Topological properties of manifolds

We now proceed to the constructions that are directly related to what we shall later use

Theorem A smooth manifold has a compact exhaustion and is paracompact

A compact exhaustion is an increasing countable collection of compact sets K_1\subset K_2\subset\dots such that M=\bigcup K_i and K_i\subset\mathrm{int}K_{i+1} for all i.

The fundamental lemma we need is a smooth version of Urysohn’s lemma

Theorem (Smooth Urysohn Lemma) If M is a smooth manifold and A,B\subset M are disjoint closed sets, then there exist a smooth function f:M\to [0,1] such that A=f^{-1}(0 and B=f^{-1}(1).

Tangent space

One can attach to every point x of a differentiable manifold a tangent space, a real vector space that intuitively contains the possible “directions” at which one can tangentially pass through x. The elements of the tangent space are called tangent vectors atx. All tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold

Definition as velocities of curves

Suppose M is a C^k manifold (k\geq 1) and x is a point in M. Pick a chart \varphi:U\to\mathbf{R}^n where U is an open subset of M containing x. Suppose two  curves \gamma_1:(-1,1)\to M and \gamma_2:(-1,1)\to M with \gamma_1(0)=\gamma_2(0)=x are given such that \varphi\circ\gamma_1 and \varphi\circ\gamma_2 are both differentiable at 0. Then \gamma_1 and \gamma_2 are called equivalent at 0 if the ordinary derivatives of \varphi\circ\gamma_1 and \varphi\circ\gamma_2 coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vector of M at x. The equivalence class of the curve \gamma is written as \gamma'(0). The tangent space of M at x, denoted by T_xM, is defined as the set of all tangent vectors, it does not depened on the choice of chart \varphi.

To define the vector space operations on T_xM, we use a chart \varphi:U\to\mathbf{R}^n and define the map d\varphi_x(\gamma'(0))=\frac{d}{dt}(\varphi\circ\gamma)(0). It turns out that this map is bijective and can thus be used to transfer the vector space operation from \mathbf{R}^n over T_xM, turning the latter into an n-dimensional real vector space. Again, one need to check that this construction does not depend on the particular char \varphi chosen, and in fact is does not.

Definition via derivations

Suppose that M is a C^\infty manifold. A real valued function f:M\to\mathbf{R} belongs to C^\infty(M) if f\circ\varphi^{-1} is infinitely differentaible for every chart \varphi:U\to \mathbf{R}. C^\infty(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

Pick a point x in M. A derivation at x is a liner map D:C^\infty(M)\to\mathbf{R} that has the property that for all f,g in C^\infty(M):

\displaystyle D(fg)=D(f)\times  g(x)+f(x)\times D(g)

modeled on the product rule of calculus. If we define addition and scalar multiplication for such derivations by

\displaystyle (D_1+D_2)(f)=D_1(f)+D_2(f)\ \text{and}\ (\lambda D)(f)=\lambda D(f)

we get a real vector space which we define as the tangent space T_xM.

The relation between the  tangent vectors defined earlier and derivatoin is as follows: if \gamma is a curve with tangent vector \gamma'(0), then the corresponding derivation D(f)=(f\circ\gamma)'(0).

\displaystyle \gamma'(0)\mapsto D_\gamma \  \text{where}\ D_\gamma(f)=\frac{d}{dt}(f\circ \gamma)\Big|_{t=0}.

Differentiable manifold

The notation of  a differentiable manifold refines that a manifold by requiring the functions that transform between charts to be differentiable. A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a liner space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

Atlas

One describes a manifold using an atlas.

Defintion An atlas for a topological spae M is a collection \{(U_\alpha,\varphi_\alpha)\} of charts on M such that \bigcup U_\alpha=M.

Tangent spaces and derivatives

Partial derivatives with respect to local coordinates

Let U\to\mathbf{R}^n be a smooth chart defined over an open set U in M. Then there are smooth real-valued functions x^1,x^2,\dots,x^n on U such that

\displaystyle \varphi(u)=(x^1(u),x^2(u),\dots,x^n(u))

for all u\in U. The functions x^1,x^2,\dots,x^n determined by the chart (U,\varphi) constitute smooth local coordinate functions defined over the domain U of the chart.

Definition Let x^1,x^2,\dots,x^n be smooth local coordinates defined over an open set U in a smooth manifold M of dimension n, and let V be the corresponding open set in \mathbf{R}^n defined such that