Curvature weighted metrics on shape space of hypersurfaces in nn-space

Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from $M$ to $\mathbb R^n$. The results of \cite{Michor118}, where mean curvature weighted metrics were studied, suggest incorporating Gau{\ss} curvature weights in the definition of the metric. This leads us to study metrics on shape space that are induced by metrics on the space of immersions of the form G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}). Here f \in \Imm(M,\R^n) is an immersion of $M$ into $\R^n$ and $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$. $\bar g$ is the standard metric on $\mathbb R^n$, $f^*\bar g$ is the induced metric on $M$, \vol(f^*\bar g) is the induced volume density and $\Phi$ is a suitable smooth function depending on the mean curvature and Gau{\ss} curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space and the conserved momenta arising from the obvious symmetries. Numerical experiments illustrate the behavior of these metrics.Comment: 12 pages 3 figure

Nonnegatively curved Euclidean submanifolds in codimension two

We provide a classification of compact Euclidean submanifolds $M^n\subset{\mathbb{R}}^{n+2}$ with nonnegative sectional curvature, for $n\ge 3$. The classification is in terms of the induced metric (including the diffeomorphism classification of the manifold), and we study the structure of the immersions as well. In particular, we provide the first known example of a nonorientable quotient $({\mathbb{S}}^{n-1}\times{\mathbb{S}}^1)/{{\mathbb{Z}}_2}\subset{\mathbb{R}}^{n+2}$ with nonnegative curvature. For the 3-dimensional case, we show that either the universal cover is isometric to ${\mathbb{S}}^2\times{\mathbb{R}}$, or $M^3$ is diffeomorphic to a lens space, and the complement of the (nonempty) set of flat points is isometric to a twisted cylinder $(N^2\times{\mathbb{R}})/{\mathbb{Z}}$. As a consequence we conclude that, if the set of flat points is not too big, there exists a unique flat totally geodesic surface in $M^3$ whose complement is the union of one or two twisted cylinders over disks.Comment: Accepted for publication in Commentarii Mathematici Helvetic