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Thread: Riemann curvature tensor

  1. #1
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    Riemann curvature tensor

    Assume <u,u>=<v,v>=1. As given here: https://en.wikipedia.org/wiki/Sectional_curvature we have this relationship between the Riemann curvature tensor and sectional curvature:

    K(u,v) = <R(u,v)v,u>/sin2​a

    where a is the angle between u and v.

    What is the intuition behind this identity?

    Also, extend u to an orthonormal basis of the the tangent space {u,...,un}.

    Then we have by above R(u,v)v = c1u+...+cnun where c1=sin2​a K(u,v). Are there other identities for the other coefficients, that is, where i>1, ci=<R(u,v)v,ui>=?

    Thanks!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Riemann curvature tensor

    What is the intuition behind this identity?


    For the denominator, you mean?
    <u,u><v,v>-<u,v>^2=|u{\rm x} v|^2=|u||v|sin^2(a).
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