黎曼流形（Riemannian manifold）是一个微分流形，其中每点p的切空间都定义了点积，而且其数值随p平滑地改变。它容许我们定义弧线长度、角度、面积、体积、曲率、函数梯度及矢量域的散度。

每个R的平滑子流形可以导出黎曼度量：把R的点积都限制于切空间内。实际上，根据纳什嵌入定理，所有黎曼流形都可以这样产生。

我们可以定义黎曼流形为和R的平滑子流形是等距同构的度量空间，等距是指其内蕴度量（intrinsic metric）和上述从R导出的度量是相同的。这对创建黎曼几何是很有用的。

黎曼流形可以定义为平滑流形，其中给出了一个切丛的正定二次形的光滑截面。它可产生度量空间：

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