What is Riemann tensor: Definition and 74 Discussions

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 measure the failure of 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.

View More On Wikipedia.org
Recent contents Top users
  1. D

    Easy way of calculating Riemann tensor?

    Homework Statement Is there any painless way of calculating the Riemann tensor? I have the metric, and finding the Christoffel symbols isn't that hard, especially if I'm given a diagonal metric. Out of 40 components, most will be zero. But how do I know how to pick the indices of...
  2. D

    Riemann tensor, Ricci tensor of a 3 sphere

    Homework Statement I have the metric of a three sphere: g_{\mu \nu} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix} Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric. Homework Equations I have all the formulas I need, and I...
  3. L

    How is the Riemann tensor proportinial to the curvature scalar?

    My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.
  4. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...
  5. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...
  6. S

    Covariant derivative of riemann tensor

    what would Rabcd;e look like in terms of it's christoffels? or Rab;c
  7. D

    Quick question to clear up some confusion on Riemann tensor and contraction

    Let's say I want to calculate the Ricci tensor, R_{bd}, in terms of the contractions of the Riemann tensor, {R^a}_{bcd}. There are two definitions of the Riemann tensor I have, one where the a is lowered and one where it is not, as above. To change between the two all that I have ever seen...
  8. F

    Contraction in the Riemann Tensor

    Hi all, I'm trying to follow through some of my notes of a GR course. The notes are working towards a specific expression and the following line appears: R^{\alpha \beta}_{\gamma \delta ; \mu} + R^{\alpha \beta}_{\delta \mu ; \gamma} + R^{\alpha \beta}_{\mu \gamma ; \delta}=0 Which by...
  9. T

    General Relativity - Riemann Tensor and Killing Vector Identity

    Homework Statement I am trying to show that for a vector field Va which satisfies V_{a;b}+V_{b;a} that V_{a;b;c}=V_eR^e_{cba} using just the below identities. Homework Equations V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}(0) R^e_{abc}+R^e_{bca}+R^e_{cab}=0 (*) V_{a;b}+V_{b;a}=0 (**) The Attempt at a...
  10. O

    Calculate Riemann tensor according to veilbein

    Homework Statement How to use veilbein to calculate Riemann tensor, Ricci tensor and Ricci scalar? (please give me the details) de^a+\omega_{~b}^a\wedge e^b=0, R_{~b}^a=d\omega_{~b}^a+\omega_{~c}^a\wedge\omega_{~b}^c. The metric is...
  11. A

    General metric with zero riemann tensor

    A metric consistent with interval: \mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right) get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor. But I don't find thus...
  12. D

    No. of field equations and components or Riemann tensor?

    no. of field equations and components or Riemann tensor?? Someone was trying to explain to me about curvature in space. From what I got from what they were saying doesn't make sense to me. I'm not sure I understand what the number of components, N, of R\alpha,\beta,\gamma,\delta when compared...
  13. A

    Riemann tensor: indipendent components

    Hi, thanks for the attention and excuse for my bad english. I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor. We have two kind of riemann tensor: type (3,1) Rikml type (4,0) Rrkml There are also some symmetry...
  14. Ranku

    Riemann tensor and flat spacetime

    When Riemann tensor = 0, spacetime is flat. Is the geometry of this flat spacetime that of special relativity?
  15. A

    Riemann tensor in normal coordinates

    This is essentially a "homework question", but I'm not looking for an explicit solution so I have posted it here. 1. Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. 2. Homework Equations Riemann tensor =...
  16. A

    Riemann tensor in normal coordinates (General Relativity)

    Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. Homework Equations Riemann tensor = (derivative of connection term) - (derivative of connection term) - (connection term)(connection term) - (connection...
  17. L

    Two Dimensional Riemann Tensor

    show that in two dimensions, the Riemann tensor takes the form R_{abcd}=R g_{a[c}g_{d]b}. i've expanded the RHS to get R g_{a[c}g_{d]b}=\frac{R}{2!} [g_{ac} g_{db} - g_{ad} g_{cb}]=\frac{1}{2} R_e{}^e [g_{ac} g_{db} - g_{ad} g_{cb}] but i can't seem to simplify it down. this is problem...
  18. L

    Understanding the Riemann Tensor and its Properties in Differential Geometry

    i need to show that R_{abc}{}^{e} g_{ed} + R_{abd}{}^{e} g_{ce}=(\nabla_a \nabla_b - \nabla_b \nabla_a) g_{cd} = 0 ok well i know that R_{abc}{}^{d} \omega_d=(\nabla_a \nabla_b - \nabla_b \nabla_a) \omega_c so i reckon that R_{abc}{}^{e} g_{ed} = (\nabla_a \nabla_b - \nabla_b \nabla_a)...
  19. W

    Robertson-Walker metric in higher dimensions (and problematic Riemann tensor)

    Hello folks, this is going to be a bit longish, but please bear with me, I'm going nuts over this. For a term paper I am working through a paper on higher dimensional spacetimes by Andrew, Bolen and Middleton. You can http://arxiv.org/abs/0708.0373" . My problem/confusion is in...
  20. L

    Lie derivative and Riemann tensor

    Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...
  21. C

    Why Is the Riemann Tensor Contracted in the Einstein Field Equations?

    why does the einstein field tensor have the riemann tensor contracted? I am confused as to what purpose it serves. I have seen an explanation that it gets rid of extra information about spacetime or something like that. and also is the Ricci scalar added to einstein tensor so that the covariant...
  22. S

    Calculating Degrees of Freedom for Riemann Tensor in D Dimensions

    How many degrees of freedom has Riemann Tensor in general D dimensions and how it can be calculated?
  23. E

    Exploring the Role of Riemann Tensor in General Relativity

    A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann...
  24. B

    Unifying Curvatures with Riemann Tensor

    I always wonder how the definitions of curvatures of curves and surfaces be unified by the Riemann Tensor symbols. For surfaces, I know R_{1,2,1,2} corresponds to the Gaussian curvature of a surface. How come R_{1,1,1,1}=0 and not corresponds to the curvature of a curve in \RE^2 or in \Re^3...
Back
Top