# Calculating a second fundamental form in the space of hermitian metrics

Let $X$ be a compact Kahler manifold and let $\mathcal M$ denote the space of hermitian metrics on $X$. We'll identify a hermitian metric with a smooth, real and positive $(1,1)$-form $\omega$. Let $\mathcal K$ be the subspace of $\mathcal M$ defined by Kahler metrics, that is, those hermitian metrics $\omega$ such that $d \omega = 0$.

It is possible to equip $\mathcal M$ with a Riemannian metric, for example by using the Hodge $L^2$ metric defined by $$G(U,V)_\omega = \int_X \langle U,V \rangle dV_\omega,$$ where the inner product inside the integral is the one defined by $\omega$ on $(1,1)$-forms on $X$. Here $U$ and $V$ are vectors tangent to $\mathcal M$, or in other words, they are smooth real $(1,1)$-forms on $X$. Despite some technical issues (the fibers of the tangent space are not complete) this metric admits a Levi-Civita connection, and a curvature tensor, and it induces a metric on $\mathcal K$ by restriction.

Q: How can we calculate the second fundamental form of $\mathcal K$ in $\mathcal M$?

It is tempting to try to do this by considering the exterior derivative as a linear map from $\mathcal M$ to the space of 3-forms on $X$, and saying that $\mathcal K$ is its fiber over $0$. If we were talking about a smooth function $f : \mathcal M \to \mathbb R$, then the second fundamental form of the fiber $f^{-1}(0) \subset \mathcal M$ (assuming smoothness) would be given by the Hessian $-\nabla^2 f$ (see Lang's "Fundamentals of differential geometry, Prop. 2.1, p. 376). Is there a similar formula when the submanifold in question is defined by a map $f : \mathcal M \to \mathcal A$ where the target space is an infinite-dimensional manifold?

An alternative approach would be to use the orthogonal projection onto the normal bundle of $\mathcal K$ in $\mathcal M$, but this projection is expressed using the Laplacian and Green operator associated to the metric $\omega$, so this road promises to be quite bumpy if at all usable. Any references or remarks would be greatly appreciated.

• Do you know this paper of Clarke-Rubinstein arxiv.org/abs/1102.3787 ? Section 3 might be of some help. Jun 13, 2012 at 14:05
• Hmm... I know a related paper of their's, the problems considered are quite similar. I'll take a closer look tomorrow, but it at a glance it seemed that their approach would be similar to the one that involvs calculating how the Laplacian and Green operator vary with the metric, which sounds painful. Jun 13, 2012 at 14:54
• I don't see how you can avoid projection onto the normal bundle. The simplest formula I know for the second fundamental form is that it's the orthogonal projection of the ambient Levi-Civita connection onto the normal bundle. In other words, if $\pi: T_* \mathcal M \rightarrow N_*\mathcal K$ is orthogonal projection onto the normal bundle, then given any two tangent vector fields $X$ and $Y$ on $\mathcal{K}$, $$II(X, Y) = \pi(\widetilde\nabla_X Y)$$ Jun 13, 2012 at 16:05