Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product $g_{p}$ $g_{p}$ on the tangent space $T_{p}M$ $T_{p}M$ at each point $p$ $p$ that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then $p\mapsto g_{p}(X(p),Y(p))$ $p\mapsto g_{p}(X(p),Y(p))$ is a smooth function. The family $g_{p}$ $g_{p}$ 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.