In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP 2m in CP n. When it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CP n (but not among all submanifolds in this Z/2 homology class). Also the totally geodesic RP 2m−1 minimizes volume in its Hamiltonian deformation class in CP n. As a...

We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that th...

In this paper, we prove that the orthogonal complement $\mathcal{F}^{\perp}$ of a totally geodesic foliation $\mathcal{F}$ on complete semi-Riemannian manifold $(M,g)$ satisfying certain inequality between mixed sectional curvatures and integrability tensor is geodesic. We also obtain conditions for existence foliations with bundle-like metric $g$.

Let M be a simply connected complete Kähler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M . Let Uν be the unit normal bundle of N in M . We prove that if a distance function ρ is differentiable at v ∈ Uν , then ρ is also differentiable at −v.

In this paper, we prove the existence of maximal slices in anti-de Sitter spaces (ADS spaces) with small boundary data at spatial infinity. The main argument is implicit function theorem. We also get a necessary and sufficient condition for boundary behavior of totally geodesic slice in ADS space. Moreover, we show that any isometric and maximal embedding of hyperbolic spaces into ADS space mus...