Posted by: cnpde | September 10, 2012

Basics of Riemann Geometry 1

Manifold is an important geometric object in modern mathematics. It is the extension of Euclid space, which is defined as a Hausdorff space locally homeomorphic to an open subset of Euclidean space  \mathbb{R}^m.  On each subset we can define chart, so that locally the manifold is exactly same with Euclidean space.

After we defined the manifold, we can define smooth functions on each subset of the manifold, however, all the smooth functions defined near a point p, under the operations + and multiplication with real numbers, cannot form a linear space, because the nul element is not defined uniquely. So we take the equivalent class, which is called C^\infty-germ of manifold at point p. All the germs form a vector space, denoted as \mathcal{F}_p.

By defining parametric curve \gamma , we can define a linear functional on  \mathcal{F}_p, <<\gamma, [f]>>= \frac{d(f\circ \gamma)}{dt} |_{t=0} . Using this notation, we can define a subspace of \mathcal{F}_p by

\\ \mathcal{H}_p=\{[f]\in \mathcal{F}_p | <<\gamma,[f]>>=0, \forall \gamma\in \Gamma_p\} ,

where \Gamma_p is the set of all parametric curve through p.

The quotient space \mathcal{F}_p/\mathcal{H}_p is the cotangent space of manifold at p, denoted as T_p^*. The element in this set is called cotangent vector, denoted as (df)_p.

Define equivalent relation in \Gamma_p as

<<\gamma,(df)_p>>=<<\gamma',(df)_p>>,

then each equivalent class [\gamma] is a linear functional on cotangent space T^*_p.  All the equivalent classes comprise tangent space T_p.

An element X in tangent space T_p can be viewed as a linear functional on C_p^\infty by denoting Xf=<X,(df)_p>, which we called directional derivative. Here we can also get another definition for tangent space T_p, which is the set of all linear operators on C_p^\infty satisfying X(fg)=f(p)\cdot Xg+g(p)\cdot Xf.

 

It is useful to express tangent vector and cotangent vector by linear combinition of natural basis.

X=\sum\limits_{i=1}^m\xi^i\frac{\partial}{\partial u^i},

a=\sum \limits_{i=1}^ma_idu^i,

where \xi^i=\frac{d(u^i\circ\gamma)}{dt}, a_i=\frac{\partial f}{\partial u^i}.

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