Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Riemann curvature tensor
Summary
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
Let (M, g) be a Riemannian or pseudo-Riemannian manifold, and be the space of all vector fields on M. We define the Riemann curvature tensor as a map by the following formula where is the Levi-Civita connection:
or equivalently
where [X, Y] is the Lie bracket of vector fields and is a commutator of differential operators. It turns out that the right-hand side actually only depends on the value of the vector fields at a given point, which is notable since the covariant derivative of a vector field also depends on the field values in a neighborhood of the point. Hence, is a -tensor field. For fixed , the linear transformation is also called the curvature transformation or endomorphism. Occasionally, the curvature tensor is defined with the opposite sign.
The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (4)
An Accelerated First-Order Method for Non-convex Optimization on Manifolds
Nicolas Boumal, Christopher Arnold Criscitiello
We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost funct
SPRINGER2022Scalar geometry and masses in Calabi-Yau string models
Claudio Scrucca, Daniel Farquet
We study the geometry of the scalar manifolds emerging in the no-scale sector of Kahler moduli and matter fields in generic Calabi-Yau string compactifications, and describe its implications on scalar
Springer2012Related people (1)
Related concepts (67)
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900.
Related courses (34)
MATH-429: Lie groups
Lie groups are manifolds with a group structure. The interaction between the geometric and the algebraic structure of these objects gives rise to a rich and beautiful subject with various applications
PHYS-427: Relativity and cosmology I
Introduce the students to general relativity and its classical tests.
Related lectures (371)
Understanding Chaos in Quantum Field Theories
Explores chaos in quantum field theories, focusing on conformal symmetry, OPE coefficients, and random matrix universality.
Structural Mechanics: Beam Bending and Boundary Conditions
Explores the moment-curvature relation for beams, emphasizing stress distribution and typical boundary conditions.
Harmonic Forms: Main Theorem
Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.
Related MOOCs (1)
Introduction to optimization on smooth manifolds: first order methods
Learn to optimize on smooth, nonlinear spaces: Join us to build your foundations (starting at "what is a manifold?") and confidently implement your first algorithm (Riemannian gradient descent).