Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Riemann curvature tensor

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The tensor measuring failure of covariant derivatives to commute and of parallel transport to be path independent.

Record metadata

Canonical name: Riemann curvature tensor
Concept ID: riemann_curvature_tensor
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Differential geometry
Domains: Differential geometry
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Curvature tensor

Carrier(s)

manifold with connection

Data

tensor $R(X,Y)Z$

Axioms / constraints

multilinearity and curvature symmetries appropriate to the connection

Canonically induces

sectional curvature
Ricci contraction for metric connections

Notes

Sign conventions for $R$ vary.

Concept sources

Wikipedia — Riemann curvature tensor — Riemann curvature tensor · encyclopedic reference · source ID wp-riemann_curvature_tensor
nLab — Riemann curvature tensor — Riemann curvature tensor · mathematical reference · source ID nlab-riemann_curvature_tensor
Cambridge — General Relativity — General Relativity · University of Cambridge lecture notes · source ID tong-gr

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Levi–Civita connection Riemann curvature tensor

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: riemann_from_connection
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: connection curvature

Authored explanation

The Levi–Civita connection determines the Riemann curvature tensor.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Levi-Civita connection — Levi-Civita connection · encyclopedic reference · source ID wp-levi_civita_connection
nLab — Levi-Civita connection — Levi-Civita connection · mathematical reference · source ID nlab-levi_civita_connection
Cambridge — General Relativity — General Relativity · University of Cambridge lecture notes · source ID tong-gr
Wikipedia — Riemann curvature tensor — Riemann curvature tensor · encyclopedic reference · source ID wp-riemann_curvature_tensor
nLab — Riemann curvature tensor — Riemann curvature tensor · mathematical reference · source ID nlab-riemann_curvature_tensor

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.