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Thread: a variant of Ricci curvature

  1. #1
    Mar 2011

    a variant of Ricci curvature

    There is a riemannian metric g on an open domain of [\mathbb{R}^n][/tex] to which we associate its Levi-Civita connexion. If we take a vector field A and as “Riemann curvature”

    [R^A_{X,Y}Z=R_{X,Y}Z+g(Y,Z)D_X^A-g(X,Z)D_Y^A+ (g(X,A)g(Y,Z)-g(Y,A)g(X,Z))A,]

    determine the Ricci curvature

    [r^A(Y,Z)=Trace(X\rightarrow R^A_{X,Y}Z)][/tex]

    and the conditions on A such that [r^A][/tex] is a 2 two times symmetric covariant tensor.

    I was thinking about taking the trace of each element of the sum by using formulas of the type,

    [r(Y,Z)=\sum_i g(R_{e_i,Y}Z,e_i)][/tex]

    but after struggling with many indices, I haven't arrived to make any simplification.

    Or maybe could it be another way to solve it?

    P.S. Could anyone tell how to introduce the mathematical text properly? I haven't found how on the site.
    Last edited by Agnete; Mar 11th 2011 at 02:36 PM.
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