7 RIEMANNIAN GEOMETRY

A manifold is a topological space which locally looks like ${\mathbb{R}}^n$ . Calculus on a manifold is assured by the existence of smooth coordinate systems. A manifold may carry a further structure if it is endowed with a metric tensor, which is a natural generalization of the inner product between two vectors in ${\mathbb{R}}^n$ to an arbitrary manifold. With this new structure, we define an inner product between two vectors in a tangent space $T_{p}M$ . We may also compare a vector at a point $p\in M$ with another vector at a different point $p^{\prime }\in M$ with the help of the ‘connection’.

There are many books about Riemannian geometry. Those which are accessible to physicists are Choquet-Bruhat et al (1982), Dodson and Poston (1977) and Hicks (1965). Lightman et al (1975) and chapter 3 of Wald (1984) are also recommended.

Example 7.3.

Exercise 7.10.

Problems

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

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