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Riemann curvature scalar, Ricci Scalar.What does they measure ?

  1. Jun 25, 2013 #1
    hello
    Can you perhaps explain what does the Riemann curvature scalar R measure? or is just an abstract entity ?
    What does the Ricci tensor measure ?
    I just want to grasp this and understand what they do.
    cheers,

    typo: What DO they measure in the title.
     
  2. jcsd
  3. Jun 25, 2013 #2

    WannabeNewton

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    Let ##M## be a Riemannian manifold and let ##p \in M##. Given any unit vector ##X\in T_p M##, pick an orthonormal basis ##\{E_{i}\}## for ##T_p M## such that ##E_1 = X##. It can be shown that ##\text{Ricci}(X,X) = \sum _{k = 2}^{n}K(E_1,E_k)## where ##K(E_1,E_i)## is the sectional curvature of the plane spanned by ##E_1 = X## and ##E_i##. So the Ricci tensor can be interpreted as the sum of sectional curvatures of planes spanned by a unit vector ##X## in the tangent space and other elements of an orthonormal basis to which ##X## belongs.

    Then, ##R = R_{j}{}{}^{j} = \sum_{j \neq k} (E_j,E_k)## i.e. the Ricci scalar measures the sum of all sectional curvatures of planes spanned by distinct pairs of elements in a given orthonormal basis.

    There are many other ways to interpret it as well. See for example: http://en.wikipedia.org/wiki/Ricci_curvature

    Also see problem 9 of chapter 4 in Do Carmo "Riemannian Geometry" for an absolutely beautiful relationship between the Ricci scalar and the Ricci tensor in terms of the area of a sphere in the tangent space and the integral of the Ricci tensor over that sphere.
     
  4. Jun 25, 2013 #3

    Nugatory

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    Staff: Mentor

    There are pretty decent intuitive definitions at math.ucr.edu/home/baez/gr/
     
  5. Jun 25, 2013 #4
    Thanks to all.
    In wikipedia it is mentioned :
    "Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space"

    Does this mean that the volume of the sphere in the curved space which is 4/3Pi*r*r*r , which is also the same in eucledian space yields a null value for the Scalar curvature ?
     
  6. Jun 25, 2013 #5

    WannabeNewton

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    Let ##\epsilon > 0## be sufficiently small so that for ##p \in M##, ##B_{\epsilon}(p)\subset M## is a geodesic ball under the exponential map. It can be shown then that ##\frac{\text{Vol}(B_{\epsilon}(p))}{\text{Vol}(B_{\epsilon}(0)\subset \mathbb{R}^{n})} = 1 - \frac{R}{6(n + 2)}\epsilon^{2}+ O(\epsilon^{4})##. If the two volumes agree then ##R = 0## to fourth order in ##\epsilon##.
     
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