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)

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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:

• (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) $\tau _{jk}=\tau _{j}\circ \tau _{k}^{-1}$

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 $x=\tau _{jk}(y)$.

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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:

• (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) $\tau _{jk}=\tau _{j}\circ \tau _{k}^{-1}$

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 $x=\tau _{jk}(y)$.

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 w: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:

$d(fg)=fdg+dfg$

$df$ 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 $Imh$ is d-closed.

'Theorem (the vanishing of the Nijenhuis tensor) The following are equivalent.

• Nijenhuis tensor vanishes.
• ${\bar {\partial }}\circ {\bar {\partial }}=0$
• $d=\partial +{\bar {\partial }}$

References

• Comparative Foundations of Differential Geometry