User:TakuyaMurata/Differentiable manifolds

Status: ''I'm putting this here because it is really in a bad shape. My hope is that this will be incorporated later into some book as a chapter when it gets more mature. -- Taku (talk) 03:30, 12 July 2008 (UTC)'' --- A topological manifold $$M$$ of real dimension $$n$$ is a paracompact separable Hausdorff topological space, where, for any $$x \in M$$, there is a neighborhood U of x and a homeomorphism $$\phi$$ such that $$\phi(U)$$ is a subset of $$R^n$$. $$\phi$$, or more precisely, the pair (\phi,

equipped with an atlas, a collection of pairs of an open subset $$U_j$$ of $$M$$ and homeomorphism $$\tau_j: U_j \to \tau_j(U_j)$$ satisfying the following conditions: In the above, $$U_j$$ is called a coordinate patch, and $$\tau_j$$ a charts on $$U_j$$. For each point $$x \in M$$, $$\tau_j(p)$$, (which is an element of $$\mathbf{R}^n$$), is called a local coordinates at $$p$$. (ii) means that local coordinates are compatible in the sense: If $$x$$ and $$y$$ are local coordinates at the same point but are given by $$\tau_j$$ and $$\tau_k$$, respectively, then $$x = \tau_{jk} (y)$$.
 * (i) $$\tau_j(U_j)$$ is an open subset of $$\mathbf{R}^n$$, and $$U_j$$ are a locally finite open cover of $$M$$.
 * (ii) $$\tau_{jk} = \tau_j \circ \tau^{-1}_k$$

A fiber bundle $$(E, B, \pi, F)$$ consists of topological spaces E, B and F, and a continuous function $$\pi: E \to B$$ such that:
 * $$F \longrightarrow E \overset{\pi}\longrightarrow B \longrightarrow 0$$

is exact. E, B and F are called the total space, base space and fiber space, respectively. Unless stated otherwise, $$B$$ is assumed to be connected.

In practice, we are usually given a pair $$(B, F)$$ and are asked to specify a pair $$(E, \pi)$$. A trivial example of a fiber bundle, simply called a trivial bundle, can be defined as follows: let $$E = B \times F$$ and define $$\pi$$ by $$\pi(x, v) = x$$. Locally, this is how a fiber bundle looks like. To give a more concrete example, take $$B$$ to be a circle and $$F$$ a line segment. The trivial bundle of $$B$$ is then just a cylinder. A Möbius strip is another different fiber bundle that is not a trivial bundle.

A manifold $$M$$ of real dimension $$n$$ is a paracompact Hausdorff second-countable topological space equipped with an atlas, a collection of pairs of an open subset $$U_j$$ of $$M$$ and homeomorphism $$\tau_j: U_j \to \tau_j(U_j)$$ satisfying the following conditions: In the above, $$U_j$$ is called a coordinate patch, and $$\tau_j$$ a charts on $$U_j$$. For each point $$x \in M$$, $$\tau_j(p)$$, (which is an element of $$\mathbf{R}^n$$), is called a local coordinates at $$p$$. (ii) means that local coordinates are compatible in the sense: If $$x$$ and $$y$$ are local coordinates at the same point but are given by $$\tau_j$$ and $$\tau_k$$, respectively, then $$x = \tau_{jk} (y)$$.
 * (i) $$\tau_j(U_j)$$ is an open subset of $$\mathbf{R}^n$$, and $$U_j$$ are a locally finite open cover of $$M$$.
 * (ii) $$\tau_{jk} = \tau_j \circ \tau^{-1}_k$$

One important thing to note is that a manifold is given with the specific dimension $$n$$. More axiomatic approach can be used to give a definition of manifold with no specific dimension.

Furthermore, $$M$$ is said to be $$C^\infty$$ differentiable if $$\tau_j$$ is a differomorphism (i.e., a differentiable bijection with differentiable inverse.) A tangent vector $$v$$ at $$p \in M$$ is a differential operator of the form:
 * $$v = \sum_j a_j {\partial \over \partial x_j}$$

(where $$x_j$$ are a local coordinates at $$p$$.) By $$T_p M$$ we mean the linear space spanned by the basis $$\partial \over \partial x_j$$. The dual space of $$T_p M$$ we denote by $$T^*_p M$$.

We now give a more abstract but direct definition of $$T^*_p M$$. Let $$I$$ be the ideal of all smooth functions in $$M$$ that vanish at $$p$$, and $$I^2$$ be the set of all functions of the form $$\sum_k f_k g_k$$ for $$f_k, g_k \in I$$. We then let $$T^*_p M = I / I^2$$. The obvious advantage of this definition is that the manifold $$M$$ need not be differentiable; in fact, the definition was motivated by an approach in algebra geometry; see Zariski tangent space for this. (Note that since the second dual of $$T_p M$$ can be identified with $$T_p M$$, this definition allows us to define $$T_p M$$ as the dual of $$T^*_p M$$.)

The cotangent bundle, denoted by $$T^*M$$, is a fiber bundle with the base space $$M$$ and the fiber space being. The elements of T^*M are called one-forms. A differential k-form or just k-form is a map from $$M$$ to $$\Omega^k \otimes T^* M$$

For a function $$f$$, a differential $$df$$ of $$f$$ at $$p$$ is defined by $$df(v) = v(f)$$ for $$v \in T_p M$$. Since $$dx_j(v) = a_j$$ when $$v = \sum_k a_k {\partial \over \partial x_k}$$, we can write more explicitly:
 * $$df = \sum_k {\partial f \over \partial x_k} dx_k$$

The map that sends $$f$$ to $$df$$, denoted by $$d$$, is called a differential map. It satisfies the usual rule of differentiation of products:
 * $$d(fg) = f dg + df g$$

$$df$$ is called a differential of $$f$$. Note that not every element of $$T^*M$$ is a differential of a function.

Let $$(M, g)$$ be an oriented $$C^\infty$$ Riemann manifold. We define the inner product
 * $$\langle f_1 \wedge ...  \wedge f_p, g_1 \wedge ...   \wedge g_p \rangle = \operatorname{det} (\langle f_j, g_k \rangle)_{j, k}$$ for $$f_j, g_k \in T^* M$$

We now give a (far-reaching) generalization of the Fundamental Theorem of Calculus.

Theorem (Stokes' formula)
 * $$\int_{\partial K} u = \int_K du$$

Proof: First assume $$u$$ has compact support. A direct computation then give the formula. For the general case, use a partition of unity.$$\square$$ (TODO: need, apparently, a much more detail.)

A hermitian manifold $$(X, h)$$ is called a Kähler manifold if $$Im h$$ is d-closed.

'Theorem (the vanishing of the Nijenhuis tensor) The following are equivalent.
 * Nijenhuis tensor vanishes.
 * $$\bar{\partial} \circ \bar{\partial} = 0$$
 * $$d = \partial + \bar{\partial}$$