English Dictionary

Riemannian manifold

名詞

(differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive definite inner product;

日本語の意味

実数体上で滑らかな微分可能多様体であり、各点の接空間に正定値な内積が定義されているもの

The geodesic equations on a Riemannian manifold determine the shortest paths between nearby points.

各点の接空間に正定値の内積が備わった滑らかな実多様体上の測地線方程式は、近くにある点同士の最短経路を決定する。

意味（1）

(differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive definite inner product;

By definition, a Riemannian manifold M has at each point p a tangent space T_pM equipped with a positive definite inner product, g_p; information about these inner products is encoded in the Riemannian metric tensor, g.

意味（2）

(more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.

Riemannian manifolds （ plural ）