A smooth manifold \(M\) equipped with Riemannian metric \(g\), denoted by \((M,g)\), is called Riemannian manifold.

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Structures on Riemannian Manifolds Link to heading

• Riemannian metric
• Levi-Civita connection
• exponential map
• Riemannian distance
• curvature

Special Riemannian Manifolds Link to heading

• Riemannian submanifold
• flat Riemannian manifold

Properties Link to heading

• geodesically complete
• metrizable

Results Link to heading

• existence of Riemannian metric