Riemannian Geometry/Bundle metrics
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Definition (bundle metric):
Let be a smooth vectur bundle. A bundle metric on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle M} is a family (), where each is a scalar product on , which satisfies the following smoothness condition: For all , the map
is smooth, ie. contained in .
Definition (Riemannian metric):
Let be a smooth manifold. A Riemannian metric is a bundle metric on the tangent space .