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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.

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Do you know this paper of Clarke-Rubinstein arxiv.org/abs/1102.3787 ? Section 3 might be of some help. –  YangMills Jun 13 '12 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. –  Gunnar Þór Magnússon Jun 13 '12 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} $, $$ II(X, Y) = \pi(\widetilde\nabla_X Y) $$ –  Deane Yang Jun 13 '12 at 16:05

1 Answer 1

Maybe, the following paper can help. Almost hermitian there means that the almost complex structure can also vay, and need not be integrable.

  • Olga Gil-Medrano, Peter W. Michor: Geodesics on spaces of almost hermitian structures, Israel J. Math. 88 (1994), 319--332. (pdf)
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