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.

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  1. I

    Problems with the Riemann tensor in general relativity

    After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative. On the internet I found equation (3), and I have tried to understand where this comes from...
  2. C

    I Ricci notations and visualisation

    I'm having trouble with notations and visualisations regarding Ricci curvature. For Riemann tensor there is variously: ##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}## ##[\nabla _{X},\nabla _{Y}]V## ##R(XY)V\mapsto Z## ##\left\langle R(XY)V,Z...
  3. binbagsss

    I First algebraic Bianchi identity of Riemann tensor (cyclic relation)

    I am guessing that: $R_{a[bcd]}=0$ can not be derived from the symmetries of $R_{ab(cd)]}=R_{(ab)cd}=0$ $R_{[ab][cd]}=0$ ?Sorry when I search the proof for it I can not find much, it tends to come up with the covariant Bianchi instead. I am guessing it will need one of the symmetries above...
  4. MrFlanders

    A GW Binary Merger: Riemann Tensor in Source & TT-Gauge

    In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
  5. G

    I Calculate Contraction 2nd & 4th Indices Riemann Tensor

    How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！
  6. physicsuniverse02

    Does anyone know which are Ricci and Riemann Tensors of FRW metric?

    I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.
  7. BiGyElLoWhAt

    I A couple questions about the Riemann Tensor, definition and convention

    According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
  8. M

    I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

    Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
  9. jk22

    I Riemann Tensor, Stoke's Theorem & Winding Number

    I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve. If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
  10. J

    A Riemann Tensor Formula in Terms of Metric & Derivatives

    Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula. Thanks.
  11. JD_PM

    I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms

    I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...
  12. V

    I Riemann Tensor: Questions & Geometric Interpretation

    Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the...
  13. saadhusayn

    Finding the Ricci tensor for the Schwarzschild metric

    I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor. The given distance element is $$ ds^2 = e^{2 \lambda} dt^2 -...
  14. Chromatic_Universe

    Specific proof of the Riemann tensor for FRW metric

    Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...
  15. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

    I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
  16. D

    I Problem: perturbation of Ricci tensor

    I am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...
  17. t_r_theta_phi

    I Intuitive explanation for Riemann tensor definition

    Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...
  18. Z

    I Components of Riemann Tensor: 4 Indexes, 16x16 Matrix

    Hello, Riemann tensor ##R^i_{jkl}## 4 indexes, and it should be matrix 16x16 in spacetime if we have time coirdinate - 0 and space coordinates -1,2,3. But how should I write the components to matrix? For example ##\begin{pmatrix}R^0_{000} & R^1_{000} & R^2_{000} ... \\ R^0_{100} & R^1_{100} &...
  19. S

    Mathematica How to compute the Riemann tensor using RGTC mathematica?

    I am trying to learn RGTC. It gives value of particular Riemann tensor but not all of it. What should I do?
  20. S

    I Number of independent components of the Riemann tensor

    I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...
  21. davidge

    I Riemann tensor in 3d Cartesian coordinates

    Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...
  22. mhob

    A Contraction between Levi-Civita symbol and Riemann tensor

    How to proof that εμνρσ Rμνρσ =0 ? Thanks.
  23. tommyxu3

    I Ricci curvatures determine Riemann curvatures in 3-dimension

    Hello~ For usual Riemann curvature tensors defined: ##R^i_{qkl},## I read in the book of differential geometry that in 3-dimensional space, Ricci curvature tensors, ##R_{ql}=R^i_{qil}## can determine Riemann curvature tensors by the following relation...
  24. T

    I How does parallel transportation relates to Rieman Manifold?

    Source: Basically the video talk about how moving from A to A'(which is basically A) in an anticlockwise manner will give a vector that is different from when the vector is originally in A in curved space. $$[(v_C-v_D)-(v_B-v_A)]$$ will equal zero in flat space...
  25. redtree

    A Riemann Tensor Equation: Simplifying the Riemann-Christoffel Tensor

    The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...
  26. A

    I What are the independent components of the Riemann tensor

    What 20 index combinations yield Riemann tensor components (that are not identically zero) from which the rest of the tensor components can be determined?
  27. redtree

    A Weyl Vacua Solutions to GR: Derivation from Riemann Tensor

    Where can I find a derivation of the vacuum solution for GR directly from the Riemann tensor of zero trace, i.e., Weyl tensor, instead of the more traditional Schwarzschild derivation?
  28. mertcan

    A Riemann tensor and covariant derivative

    hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...
  29. W

    Riemann tensor given the space/metric

    Homework Statement Given two spaces described by ##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv## ##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv## Calculate the Riemann tensor Homework Equations Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...## We have...
  30. D

    Covariant derivative of Killing vector and Riemann Tensor

    I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...
  31. C

    Calculating Covariant Riemann Tensor with Diag Metric gab

    Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...
  32. ShayanJ

    Contracting Riemann tensor with itself

    In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzschild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its...
  33. D

    Proving Non-linear Wave Equation for Riemann Tensor

    Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...
  34. D

    Deriving Riemann Tensor Comp. in General Frame

    How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
  35. binbagsss

    R computation from 1 independent Riemann tensor component

    We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components. The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## . I thought it would have been...
  36. Einj

    Riemann tensor and derivatives of ##g_{\mu\nu}##

    Hello everyone, I'm studying Weinberg's 'Gravitation and Cosmology'. In particular, in the 'Curvature' chapter it says that the Riemann tensor cannot depend on ##g_{\mu\nu}## and its first derivatives only since: What I don't understand is how introducing the second derivatives should change...
  37. aditya ver.2.0

    The modelling of space time through Riemann tensor calculus

    How do we explains space-time through Riemann Calculus?
  38. S

    Symmetry of Riemann Tensor: Investigating Rabmv

    We know how objects such as the metric tensor and the Cristoffel symbol have symmetry to them (which is why g12 = g21 or \Gamma112 = \Gamma121) Well I was wondering if the Riemann tensor Rabmv had any such symmetry. Are there any two or more particular indices that I could interchange and...
  39. Mr-R

    I Calculating the Riemann Tensor for a 4D Sphere

    Dear All, I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed} But the exercise asks me to calculate R_{abcd}. Do...
  40. C

    Projection of the Riemann tensor formula.

    Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V. I'm trying to prove the formula $$R(X',Y'')Z' \cdot V'' = (Z' \cdot (\nabla'_{X'}B') + \left<X'\cdot B', Z' \cdot B'\right>)(Y'', V'') + (V''...
  41. V

    Using parallel propagator to derive Riemann tensor in Sean Carroll's

    Hello all, In Carroll's there is a brief mention of how to get an idea about the curvature tensor using two infinitesimal vectors. Exercise 7 in Chapter 3 asks to compute the components of Riemann tensor by using the series expression for the parallel propagator. Can anyone please provide a...
  42. P

    Why Do These Riemann Tensor Terms Cancel Each Other Out?

    I was working on the derivation of the riemann tensor and got this (1) ##\Gamma^{\lambda}_{\ \alpha\mu} \partial_\beta A_\lambda## and this (2) ##\Gamma^{\lambda}_{\ \beta\mu} \partial_\alpha A_\lambda## How do I see that they cancel (1 - 2)? ##\Gamma^{\lambda}_{\ \alpha\mu}...
  43. S

    What is the Form of Riemann Tensor in 3D?

    hi Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it. see for example Geometry,Topology and physics By Nakahara Ch.7 In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel...
  44. Z

    Calculate the elements of the Riemann tensor

    Homework Statement Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries) Homework Equations R_{abcd}=R_{cdab} R_{abcd}=-R_{abdc} R_{abcd}=-R_{bacd} The Attempt at a Solution I don't see how 21...
  45. N

    Local flat space and the Riemann tensor

    Can somebody explain in layman's terms the connection between local flat space (tangent planes) on a manifold and the Riemann tensor.
  46. M

    Riemann tensor cyclic identity (first Bianchi) and noncoordinate basis

    I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...
  47. S

    Calculation of double dual of Riemann tensor

    Hi all, I encounter a technical problem about tensor calculation when studying general relativity. I think it should be proper to post it here. Riemann curvature tensor has Bianchi identity: R^\alpha_{[\beta\gamma\delta;\epsilon]}=0 Now given double (Hodge)dual of Riemann tensor: G = *R*, in...
  48. I

    Interpretation of torsion vs riemann tensor

    Hi all, I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner. It appears from "Geometric...
  49. H

    Calculating the Riemann Tensor

    The work I have been following has me very confused... and I am almost sure I am making a mistake somewhere! After working up to this equation: \delta V = dX^{\mu}\delta X^{\nu} [\nabla_{\mu} \nabla_{\nu}]V I am asked to calculate the curvature tensor. Now the way I did it, turned out...
  50. O

    No. of Independent Components of Riemann Tensor in Schwartzchild Metric

    In general 4d space time, the Riemann tensor has 20 independent components. However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have? (I think it is 4, but I...
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