Riemannian manifolds

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product ${\displaystyle g_{p}}$ on the tangent space ${\displaystyle T_{p}M}$ at each point ${\displaystyle p}$ that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then ${\displaystyle p\mapsto g_{p}(X(p),Y(p))}$ is a smooth function. The family ${\displaystyle g_{p}}$ of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

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