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On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge dx^\nu$, consider the following term. $$\int_M tr(R\wedge *R)=2\int_M tr(R_{\mu\nu}R^{\mu\nu})dvol_g=2\int_M tr(R_{\mu\nu}{}_{\rho\sigma}R^{\mu\nu\sigma\rho})dvol_g$$ This is Yang-Mills action evaluated on Levi-Civita. My questions are

What is its meaning? Is it a topological term?

Does it have any physical applications, especially in GR?

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    $\begingroup$ If you google "L2 norm of curvature", you'll find some papers regarding this functional, especially in dimension 4. $\endgroup$
    – Deane Yang
    Commented Mar 28, 2014 at 21:52

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You can find information about functionals that are quadratic in the curvature in Besse's book Einstein Manifolds. In particular, see Chapter 4, Section H and the references cited therein. Your questions about topological interpretations and possible connections with GR, etc., are addressed in some of those references.

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