59 research outputs found

### On submanifolds whose tubular hypersurfaces have constant mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds
whose tubular hypersurfaces have some constant "higher order mean curvatures".
Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is
defined as the $k$-th power sum of the principal curvatures, or equivalently,
of the shape operator. Many necessary restrictions involving principal
curvatures, higher order mean curvatures and Jacobi operators on such
submanifolds are obtained, which, among other things, generalize some classical
results in the theory of isoparametric hypersurfaces given by E. Cartan, K.
Nomizu, H. F. M{\"u}nzner, Q. M. Wang, \emph{etc.}. As an application, we
finally get a geometrical filtration for the focal varieties of isoparametric
functions on a complete Riemannian manifold.Comment: 29 page

### Differentiable classification of 4-manifolds with singular Riemannian foliations

In this paper, we first prove that any closed simply connected 4-manifold
that admits a decomposition into two disk bundles of rank greater than 1 is
diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$,
$\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm
\mathbb{CP}^2$. As an application we prove that any closed simply connected
4-manifold admitting a nontrivial singular Riemannian foliation is
diffeomorphic to a connected sum of copies of standard $\mathbb{S}^4$,
$\pm\mathbb{CP}^2$ and $\mathbb{S}^2\times\mathbb{S}^2$. A classification of
singular Riemannian foliations of codimension 1 on all closed simply connected
4-manifolds is obtained as a byproduct. In particular, there are exactly 3
non-homogeneous singular Riemannian foliations of codimension 1, complementing
the list of cohomogeneity one 4-manifolds.Comment: 24 pages, final version, to appear in Math. An

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