442

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Course Name:	442 Differential Geometry and General Relativity
Lecturer:	Dr. Calin Lazaroiu
Course Description:	http://www.maths.tcd.ie/pub/official/Courses06-07/442.html
Lecturer's Page:	http://www.maths.tcd.ie/~calin/teaching/442.html

Manifolds

Definitions

Manifold: A manifold is a topological space that is locally Euclidean i.e. around every point there is a neighbourhood that can be mapped homeomorphically to the open unit ball in $\reals^n$

Chart: A chart $(U,h)$ of dimension n on $(X,\tau)$ is a bijective map $h:U\rightarrow V$ where

$U\subset X$ is a non-void open subset of $X$ .
$V\subset\reals^n$ is an open subset of $\reals^n$ .
$h$ is homeomorphic from $V$ to $U$ .

Smooth Atlas: A smooth atlas on $(X,\tau)$ is a family $\mathcal A=\{(U_\alpha ,h_\alpha)| \alpha \in A\}$ of charts $(U_\alpha ,h_\alpha)$ such that

$X=\bigcup_{\alpha\in A} U_\alpha$ .
$\forall \alpha ,\beta \in A$ the map (known as the transition map) $h_\beta \circ {h_\alpha}^{-1} = h_\beta |_{U_\alpha \cap U_\beta} \circ h^{-1}|^{U_\alpha \cap U_\beta}:h_\alpha(U_\alpha \cap U_\beta) \rightarrow h_\beta (U_\alpha \cap U_\beta)$ is infinitely differentiable.

Compatibility: Two atlases $\mathcal A=\{(U_\alpha ,h_\alpha)| \alpha \in A\}$ , $\mathcal B=\{(V_\beta ,k_\beta)| \beta \in B\}$ are compatible if $\mathcal A \cup \mathcal B$ is again an atlas.

here are some of Eoin Curran's Notes

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