On Ricci curvatures of hypersurfaces in abstract Wiener spaces (Q1911534)

Let \((B, H, \mu)\) be an abstract Wiener space, where \(B\) is a real separable Banach space, \(H\) is a separable Hilbert space with norm \(|\cdot |_H\), which is embedded continuously and densely in \(B\), and \(\mu\) is a probability measure on \(B\) such that \[ \int_B \exp [\sqrt{-1} \langle z,\ell\rangle]\mu(dz) = \exp [-|\ell|^2_H/2],\quad \ell \in B^*. \] According to Getzler, Airault, van Biesen, Kazumi, and Shigekawa, for every nondegenerate infinitely differentiable function \(G : B \to \mathbb{R}^d\), for the submanifold \(M = \{G = b\}\) can be defined the Ricci curvature. The author considers the case of hypersurfaces \((d = 1\) above) and constructs examples showing that the Ricci curvature is no longer a geometrical quantity. In particular, it is shown that for ``elliptic'' hypersurfaces of the type \[ S =\Bigl\{\sum^\infty_{k=1} a_k \langle \cdot, \ell_k\rangle^2 = a\Bigr\} \] the sign of the Ricci curvature changes as \(a\) varies.