Riemann Curvature TensorEdit

I can’t write this from a political viewpoint, but I can provide a clear, neutral encyclopedia-style article on the Riemann curvature tensor with appropriate intra-article links.

The Riemann curvature tensor is a central object in differential geometry that encodes how a smooth manifold bends and twists. It captures how parallel transport around infinitesimal loops fails to be trivial, and it serves as a precise measure of curvature in any dimension. In physics, especially in general relativity, the Riemann tensor describes how spacetime itself is curved by matter and energy. For a concise introduction and the standard notation, see the basic definitions below and the linked terms throughout.

Definition

Let M be a smooth manifold equipped with a symmetric, nondegenerate metric g (a metric tensor), which induces a unique torsion-free connection known as the Levi-Civita connection. The components of the connection are the Christoffel symbols Γ^ρ{ μ ν}. The Riemann curvature tensor is a (1,3)-tensor whose mixed components are defined by R^ρ{ σ μ ν} = ∂μ Γ^ρ{ σ ν} − ∂ν Γ^ρ{ σ μ} + Γ^ρ{ λ μ} Γ^λ{ σ ν} − Γ^ρ{ λ ν} Γ^λ{ σ μ}.

Equivalently, it can be viewed as the failure of second covariant derivatives to commute: R^ρ{ σ μ ν} v^σ = (∇μ ∇ν − ∇ν ∇_μ) v^ρ for any vector field v, where ∇ denotes the covariant derivative compatible with g.

In coordinate components, the Riemann tensor has the symmetries - R^ρ{ σ μ ν} = − R^ρ{ σ ν μ} (antisymmetric in μ, ν), - R^ρ{ σ μ ν} = − R{ σ ρ μ ν} (antisymmetric in ρ, σ after lowering the index), - R^ρ{ σ μ ν} = R{ μ ν σ}^ρ (pair symmetry), and the first Bianchi identity R^ρ_{ [ σ μ ν] } = 0, expressed as a cyclic sum over μ, ν, σ.

The tensor with all indices lowered, R_{ρ σ μ ν}, shares the same symmetries and provides a useful representation in many computations.

Basic contractions

The Riemann tensor contains within it several other curvature quantities obtained by contraction: - The Ricci tensor: R_{ σ ν } = R^ρ{ σ ρ ν }. - The scalar curvature: R = g^{ σ ν } R{ σ ν }. These contractions summarize certain aspects of curvature and play a central role in the Einstein field equations of general relativity.

There is also a decomposition of the full curvature in dimensions n ≥ 3 into trace and trace-free parts, notably involving the Weyl tensor and the Ricci part.

Geometric interpretation

Geodesic deviation: The Riemann tensor governs how nearby geodesics separate or converge. The geodesic deviation equation expresses the relative acceleration of nearby free-falling particles in terms of R^μ_{ ν α β} u^ν u^α ξ^β, where u is the tangent vector to a geodesic and ξ is the separation vector between neighboring geodesics.
Local curvature measures: Curvature describes the intrinsic bending of the manifold that cannot be removed by any smooth change of coordinates. The Riemann tensor is the precise mathematical object that encodes this intrinsic bending.

Examples

Flat space: In Euclidean space with the standard metric, all Christoffel symbols vanish in Cartesian coordinates, and the Riemann tensor is identically zero, reflecting zero intrinsic curvature.
Sphere: For a sphere of radius a, the intrinsic curvature is constant and positive. In a coordinate system adapted to the sphere, the nonzero components satisfy R_{θ φ θ φ} = a^2 sin^2 θ, reflecting positive curvature K = 1/a^2.
Hyperbolic space: Hyperbolic space has constant negative curvature; its Riemann tensor components reflect K = −1/a^2, with a coordinate-dependent expression that can be written in terms of the metric and the constant curvature.

Computational aspects

Coordinate expressions: To compute R^ρ_{ σ μ ν}, one first computes the Christoffel symbols from the metric g, then applies the defining formula. This is the standard route in differential geometry and is implemented in many mathematical and physical software packages.
Orthonormal frames: In an orthonormal frame, the structure coefficients (or spin connection) replace Christoffel symbols, and the Riemann tensor components can be obtained via the curvature 2-form.
Symmetries: The symmetries of the Riemann tensor reduce the number of independent components, and the Bianchi identities impose differential relations among them.

Connections to other topics

Relationship to the metric: The Levi-Civita connection is metric-compatible (∇ g = 0) and torsion-free, and it is uniquely determined by the metric. The Riemann tensor is constructed from this connection.
Tensors built from curvature: The Ricci tensor and scalar curvature are contractions of the Riemann tensor, while the Weyl tensor captures the conformal, trace-free part of curvature in dimensions n ≥ 4.
In physics: In general relativity, the Einstein tensor G_{ μ ν } is a particular contraction of the Riemann tensor (via the Ricci tensor and scalar curvature) and appears in the Einstein field equations that relate spacetime curvature to matter and energy.

Historical context

The mathematical theory of curvature traces back to the 19th century with the work of Bernhard Riemann and the development of differential geometry, later enriched by contributions from Gregorio Ricci-Curbastro and Tullio Levi-Civita in the formalization of the connection and curvature concepts. The modern language of the Riemann curvature tensor emerges from these developments and remains a foundational element in both pure mathematics and physics.