“Riemannian manifold”的概念、定义、翻译、参考文献-科学参考

Riemannian manifold

Mathematics

单词	Riemannian manifold
释义	Riemannian manifold Mathematics A manifold with a metric structure, that is, a first fundamental form, is associated with each set of local coordinates. This means intrinsic properties such as length, angle, area, and Gaussian curvature may be defined. In two dimensions, the first fundamental form is commonly written as $d s^{2} = E d u^{2} + 2 F d u d v + G d v^{2}$ in terms of local coordinates u and v. The length L of a curve (u(t),v(t)) and the area A of a region R are then defined as $\begin{array}{c} L = \int \sqrt{E {(\frac{d u}{d t})}^{2} + 2 F \frac{d u}{d t} \frac{d v}{d t} + G {(\frac{d v}{d t})}^{2}} d t, \\ A = \iint_{R} \sqrt{E G - F^{2}} d u d v . \end{array}$ If the manifold uses more than one set of coordinates, then transition maps are necessarily isometries so that the metric structure is consistent across the manifold.
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