##  [Riemannian Manifold](/riemannian-manifold-0) 

 Definition

A differentiable manifold equipped with a smoothly varying positive-definite inner product on each tangent space, which defines lengths of tangent vectors and induces notions of distance and angle on the manifold.

 

 

 

 

 

 





## Principle

Principle

Assign to every point a symmetric, bilinear, positive-definite form on the tangent space that depends smoothly on the point; this Riemannian metric organizes local linear geometry and supports metric constructions and curvature.

 

 

 

 

 





## Demonstration

Demonstration

The 2-sphere embedded in Euclidean 3-space with the metric induced from the ambient dot product: each tangent plane inherits the restriction of the Euclidean inner product, giving great-circle distances as geodesic lengths.

 

 

 

 

## Misapplication

Misapplication

Calling a manifold Riemannian when the pointwise bilinear form is indefinite (signature not all positive) — such a manifold is pseudo-Riemannian, not Riemannian — or using a merely continuous, non-smooth form where smoothness is required for standard differential constructions.

 

 

 

 

 





## Consequence

Consequence

One obtains well-defined local lengths, angles, a Levi-Civita connection, geodesics, curvature tensors, and a distance function compatible with the manifold topology; analysis and geometry (Laplacian, volume form) follow from the metric.

 

 

 

 

## Reversal

Reversal

Dropping positive-definiteness yields pseudo-Riemannian geometry with causal structure; dropping smoothness yields metric notions without differential-geometric tools; considering only the topological manifold removes metric-dependent invariants like curvature.

 

 

 

 

 





## Boundary

Boundary

Applies to smooth (C^k, usually C^∞) manifolds with a positive-definite Riemannian metric; excludes indefinite-metric manifolds, purely topological manifolds without a chosen metric, and discrete metric spaces.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with the abstract notion of metric space: not every metric on a manifold arises from a Riemannian metric; tension with 'Riemann curvature' (a derived tensor) and with pseudo-Riemannian structures used in relativity.

 

 

 

 

 





## Synthesis

Synthesis

A Riemannian manifold is a smooth manifold together with a smoothly varying positive-definite inner product on each tangent space, providing the local linear metric data that underlies lengths, angles, geodesics and curvature.