## Metric deformation:

Let $(M,g_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g_0$, $$g_t=g_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric (0,2)-tensor. A natural (and important) question is how the sectional curvatures of $g_0$ change under this deformation, e.g., what is the infinitesimal change in terms of $h$. More precisely, given two $g_0$-orthonormal vectors $X$ and $Y$ in $T_pM$, define the (unnormalized) sectional curvature $$k(t)=g_t(R_t(X,Y)Y,X),$$ where we are using the appropriate sign convention on $R$.

Q:What is the explicit formula for $k'(0)=\frac{d}{dt}k(t)\big|_{t=0}$?

## Possible (but different?) answers:

I have found a few papers with an answer, but (understandably) none provide the complete argument. Unfortunately, it seems like some of them are *really different*, and it would be very helpful if someone could point out if they coincide for some (possibly silly) reason I am not seeing.

**Berger'66 (Trois remarques sur les variétés riemanniennes à courbure positive)/Bourguignon, Deschamps, Sentenac'72 (Conjecture de H. Hopf sur les produits de variétés):**$$k'(0)=\nabla_X\nabla_Y h(X,Y)-\tfrac12\nabla_X\nabla_X h(Y,Y)-\tfrac12\nabla_Y\nabla_Y h(X,X)$$**Strake'87 (Curvature increasing metric variations)**: $$k'(0)=\nabla_X\nabla_Y h(X,Y)-\tfrac12\nabla_X\nabla_X h(Y,Y)-\tfrac12\nabla_Y\nabla_Y h(X,X)+h(R_0(X,Y)Y,X)$$

$$-k(0)(h(X,X)+h(Y,Y))$$

`3.`

**Topping'06 (Lectures on Ricci Flow):** $$k'(0)=\nabla_X\nabla_Y h(X,Y)-\tfrac12\nabla_X\nabla_X h(Y,Y)-\tfrac12\nabla_Y\nabla_Y h(X,X)+\tfrac12h(R_0(X,Y)Y,X)$$

$$-\tfrac{1}{2}h(R_0(X,Y)X,Y)$$

EDIT: I should point out that, although formula 3) as I wrote above is NOT a correct expression for $k'(0)$, I (embarrassingly) misinterpreted it from Prop. 2.3.5 in Topping's lecture notes -- which actually **contains the correct formula (matching Vitali's answer below)**. He gives a general expression for $\frac{d}{dt}g_t(R^t(X,Y)Z,W)\big|_{t=0}$, and by using the Ricci identity it becomes clear that his formula is indeed the same as Vitali's. I sincerely apologize for the confusion.

Note that all answers above coincide if $(M,g_0)$ has non-negative sectional curvature and $X$ and $Y$ span a plane of zero $g_0$-curvature. As Strake remarks, the self-adjoint endomorphism $A_X Z=R(Z,X)X$ is positive-semidefinite and $g(A_X Y,Y)=0$. (Nevertheless, to the best of my understanding, THESE HYPOTHESES ARE NOT ASSUMED in the references in 1). Also, it seems to me that answers 2 and 3 DO NOT COINCIDE.