Let Mn+p p (c) be n + p-dimensional connected semi-Riemannian manifold of constant curvature c whose index is p. It is called indefinite space form of index p. Let M be an n-dimensional Riemannian manifold immersed in Mn+p p (c). The semi-Riemannian metric of Mn+p p (c) induces the Riemannian metric of M , M is called a spacelike submanifold. Spacelike submanifolds in indefinite space form Mn+p...

Let U be a real form of a complex semisimple Lie group, and τ , σ, a pair of commuting involutions on U . This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space U/L on...

where H(x, t) is the mean curvature vector of Ft(M) and Ft(x) = F (x, t). We call F : M × [0, T ) → F(c) the mean curvature flow with initial value F . The mean curvature flow was proposed by Mullins [17] to describe the formation of grain boundaries in annealing metals. In [3], Brakke introduced the motion of a submanifold by its mean curvature in arbitrary codimension and constructed a genera...

In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift to SLag submanifolds of M , invariant under the torus action. If k = n− 1 and H...

Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds i...

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on “convenient” vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree fo...

A theorem of Lawson and Simons[8] states that the only stable minimal submanifolds in CP are complex submanifolds. We generalize their result to the cases of HP and OP. The treatment is given in the context of Jordan algebra, so that the seemingly unrelated case of S is uni…ed naturally with the above projective spaces. 1 Introduction Complex geometry is a very rich subject. Some of its beautif...

Introduction. To a smooth manifold M one can associate in a natural way a new smooth manifold, the manifold of k-jets of n-dimensional submanifolds of M , indicated by G n (M), which parametrizes in a smooth way the k-jets of immersed submanifolds ofM . OnG n (M) one can build in a canonical way a differential ideal, denoted . The cohomology associated to the complexG n (M)/ (k) is called chara...

In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \ 1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is trace-free second fundamental form. We obtain some Simons' type integral inequalities rigidity theorems for such submanifolds. Moreover $\mathbb{C}^n$ $\nabla^*\nabla^*T=0$ introduce a...

We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs.