View source for Riemannian Geometry/Bundle metrics

{{definition|bundle metric|Let <math>E \to M</math> be a smooth vectur bundle. A '''bundle metric''' on <math>M</math> is a family <math>\langle \cdot, \cdot \rangle_p</math> (<math>p \in M</math>), where each <math>\langle \cdot, \cdot \rangle_p</math> is a scalar product on <math>E_p</math>, which satisfies the following smoothness condition: For all <math>\sigma, \tau \in \Gamma(E)</math>, the map
:<math>p \mapsto \langle \sigma(p), \tau(p) \rangle_p</math>
is smooth, ie. contained in <math>\mathcal C^\infty(M)</math>.}}

{{definition|Riemannian metric|Let <math>M</math> be a smooth manifold. A '''Riemannian metric''' is a bundle metric on the tangent space <math>TM</math>.}}

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