# What is Curvature tensor: Definition and 50 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.

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1. ### B Attempted proof of the Contracted Bianchi Identity

My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...
2. ### I Reconciling 2 expressions for Riemann curvature tensor

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...
3. ### SH2372 General Relativity - Lecture 4

0:00 The metric tensor 12:55 Curve lengths 28:17 Metric compatibility of connections 35:47 The Levi-Civita connection 40:27 Induced metrics 50:12 Curvature and the metric 1:04:18 Killing fields and symmetries
4. ### A Anti-self Dual Part (2,2) Riemann Curvature Tensor

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.
5. ### I Riemann Curvature Tensor on 2D Sphere: Surprising Results

I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...
6. ### A Curvature Tensor for Dual Vectors

Good day all. Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the...
7. ### Riemann curvature Tensor

I derived this equation $$A_{i,jk}-A_{i,kj}=R^r _{kij}A_r$$.But where do I use this $$A_{i,j}+A_{j,i}=0$$?
8. ### Riemann Curvature Tensor in 2D

Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...
9. ### Proving the symmetry property of Riemann curvature tensor

Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$R_{abcd} = -R_{bacd} = -R_{abdc}$$? I'm only able to prove it with the anti-symmetry property and...
10. ### I Riemann curvature tensor derivation

Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...
11. P

### I Non-zero components of Riemann curvature tensor with Schwarzschild metric

I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...
12. ### A On the dependence of the curvature tensor on the metric

Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...
13. ### Find the Riemannian curvature tensor component

Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...
14. ### A Curvature tensor in all flat space coordinates

hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero...
15. ### A Riemann Curvature Tensor

Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space Thank You
16. ### A Can geodesic deviation be zero while curvature tensor is not

I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...
17. ### Varying The Gibbons-Hawking Term

The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##. I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##. The answer (given in eqns 6&7 of...
18. ### Checking derivation of the curvature tensor

Homework Statement I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks Homework Equations I am trying to derive the curvature...
19. ### Learning to Simplify the Curvature Tensor

I just watched susskind video on EFE but he didnt show us how to convert curvature tensor(the one with 4 indices) to that of Ricci tensor. Can anyone help me with this? Try to simplify it as I just started this.
20. ### Where should I ask about mathematical problems with Riemann curvature tensor

I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!
21. ### Third rank curvature tensor?

Hi, I am curious about expressing the curvature of a manifold using a third-rank tensor. The fourth order Riemann tensor can be contracted to give the second order Ricci tensor and zeroth order Ricci scalar, but is there no way of obtaining a third- or first-order tensors, or would this simply...
22. ### Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...
23. ### Understanding the Ricci Curvature Tensor in Einstein's Field Equations

I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...
24. ### Curvature Tensor: Non-Zero in Local Inertial Frame

hi In a local inertial frame with g_{ij}=\eta_{ij} and \Gamma^i_{jk}=0. why in such a frame, curvature tensor isn't zero? curvature tensor is made of metric,first and second derivative of metric.
25. ### What is the Riemann Curvature Tensor for Flat and Minkowski Space?

Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...
26. ### Second derivative of a metric and the Riemann curvature tensor

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...
27. ### Double contraction of curvature tensor -> Ricci scalar times metric

Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...
28. ### Calculating Riemann Curvature Tensor: Faster Methods?

I've been trying to calculate the Riemann Curvature Tensor for a certain manifold in 3-dimensional Euclidean Space using Christoffel Symbols of the second kind, and so far everything has gone well however... It is extremely tedious and takes a very long time; there is also a high probability...
29. ### EFE: Why is there a curvature tensor and curvature scalar?

In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?
30. ### Riemann's curvature tensor help

Please, anyone tell me how to proof this equation: {R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} Given a definition...
31. ### Understanding curvature tensor equation

hi, I am trying to understand the meaning of the following equation in the simplest way possible thanks in advance
32. ### Understanding curvature tensor equation

hi, I am trying to understand the meaning of the following equation in the simplest way possible https://public.blu.livefilestore.com/y1pWwu86vlTmLHRY35RBhm3I55eYrMWCtWPmdVjAM807ltH2EfInsaFIBk6nCFhnIdwno9Mz4Oa4qWC8Zv9xND3KA/tensorp.png?psid=1 [Broken] thanks in advance
33. ### Orthonormal basis => vanishing Riemann curvature tensor

Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...
34. ### Properties of curvature tensor in 3 dimensions?

Is there any properties with the curvature tensors in 3 dimensions? (Maybe between the Ricci tensor and the Ricci scalar, they are proportional to each other? ) I heard about it in a lecture, but I can not remember the details. The 3 dimensional case is not discussed in many reference books...
35. ### (anti)-symmetries of the Riemann curvature tensor

The Riemannian curvature tensor has the following symmetries: (a) Rijkl=-Rjikl (b) Rijkl=-Rijlk (c) Rijkl=Rklij (d) Rijkl+Rjkil+Rkijl=0 This is surely trivial, but I do not see how to prove that Rijkl=-Rjilk. :( Thanks.
36. ### Proove interchange symmetry of the Riemann curvature tensor

Homework Statement Proove that: R_{abcd} = R_{cdab} Homework EquationsThe Attempt at a Solution I'm not sure whether to expand the following equations any further (using the definitions for the christoffel symbols) and hope that I can re-label repeated indexes at a later stage or if there is...
37. ### Confusion wiht Curvature Tensor

I Have a problem understanding that vanishing of the curvature tensor implies that parallel transport is independent of path. With the converse of this assertion I have no problem. The text I'm reading(Lovelock and Rund) explains the converse but treats the direct assertion as trivial. Can...
38. ### Curvature tensor of sphere radius R

hello! I need to find curvature tensor of sphere of R radius. How can I start? thanks!
39. ### Riemann Curvature Tensor

(i) show that R_{abcd}+R_{cdab} (ii) In n dimensions the Riemann tensor has n^4 components. However, on account of the symmetries R_{abc}^d=-R_{bac}^d R_{[abc]}^d=0 R_{abcd}+-R_{abdc} not all of these components are independent. Show that the number of independent components is...
40. ### Equation with Riemann curvature tensor

Can anyone prove the following formula: R_{abf}^{\phantom{abf}e} \Gamma_{cd}^f = R_{abc}^{\phantom{abc}f} \Gamma_{fd}^e + R_{abd}^{\phantom{abd}f} \Gamma_{cf}^e I found it in "General Relativity" by Wald (in slightly different notation).
41. ### Degrees of freedom in the curvature tensor

The Einstein field equations (EFE) in 4 dimensions have 10 degrees of freedom; The Riemannian curvature tensor in 4 dimensions has 20. If I understood this correctly, one can split up the curvature tensor and describe the remaining degrees of freedom by its traceless part, which is called the...
42. ### Riemann Curvature Tensor

http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html" is a derivation of the components of the riemann curvature tensor. the problem is that i can't understand the transition between eq97 and eq89 . what does "To lowest order " mean ?
43. ### Riemann curvature tensor as second derivative of the metric

It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...
44. ### Unraveling the Mysteries of the Riemann Curvature Tensor

Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...
45. ### Determination of Riemann curvature tensor from tidal forces

Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...
46. ### How can we tell if a given tensor is a curvature tensor?

Under what circumstances do we know whether a given tensor of 4th rank could be the curvature tensor of a manifold? For instance, if I specify some arbitrary functions R_{ijkl} (with the necessary symmetries under interchange i<->j, k<->l, and ij<->kl), is it necessarily the case that there is a...
47. ### Energy-mom tensor does not determine curvature tensor uniquely ?

Energy-mom tensor does not determine curvature tensor uniquely ? If the energy momentum tensor is known, that fixes the Einstein tensor uniquely from the Einstein eqs. Einstein tensor is built from Riemann contractions so it doesn't fix Riemann uniquely. Does that mean a single energy momentum...
48. ### Question involving curvature tensor

Greetings, I'm working out some of the mathematical relations between Yang-Mill theory and GR. I'm having difficulty working out a somewhat trivial thing, I was wondering if anyone here could help me. To keep things concrete, I'll stick to the case D=4, but I'd like to be able to generalise to...
49. ### Find # of Independent Components of Riemann Curvature in D Dimensions

How do i find the number of independent components of the Riemann curvature tensor in D space-time dimensions. One is given that the Riemann tensor is an (2,2) irreducible rep of GL(4, \mathbb{R}) and obeys Bianchi I R_{[\mu\nu|\rho]\lambda}=0 Been trying this problem for 3 days and...
50. ### Riemann curvature tensor derivation

Hey, when calculating the Riemann curvature tensor, you need to calculate the commutator of some vector field V , ie like this :- [\bigtriangledown_a, \bigtriangledown_b] = \bigtriangledown_a\bigtriangledown_b - \bigtriangledown_b\bigtriangledown_a = V;_a_b - V;_b_a But...