It is well known that the sphere $S^6(1)$ admits an almost complex structure $J$ which nearly K\"{a}hler. A submanifold $M$ of Hermitian manifold called a CR if it differentiable distribution $\mathcal{D}_1$ such its orthogonal complement totally real distribution. In this case normal bundle also splits into two distributions $\mathcal{D}_3$, complex, and complement. proper three-dimensional si...

We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a curve geodesic ball B of the round 6-sphere meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, obtain rigidity results reflection-invariant $$\mathbb {S}^6$$ and associative cones {R}^7$$ . Second, consider with boundary Lagrangian submanifold strict 6-man...

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Rie...

We estimate the $L^2$ norm of restriction to a totally geodesic submanifold eigenfunctions Laplace-Beltrami operator on standard flat torus $\mathbb{T}^d$, $d\ge2$. reduce getting correct bounds counting lattice points in intersection some $\nu$-transverse bands sphere. Moreover, we prove for rational submanifolds arbitrary codimension. In particular, verify conjecture Bourgain-Rudnick $L^2$-re...

We introduce the geometry of pseudo-totally umbilical lightlike submanifold $M$ a semi-Riemannian manifold $\bar{M}$. In line with above, we give complete classification $1$-lightlike submanifolds, such as hypersurfaces and half-lightlike submanifolds. Furthermore, screen distributions are also investigated, for any whose umbilical. Closely linked to above show, under some geometric conditions,...

We establish the existence of non-constant closed geodesics on moduli spaces of SU(2) monopoles of arbitrary charge. More generally, we show that the moduli space of strongly centred monopoles of charge k, k 2, contains a totally geodesic submanifold which can be identiied with the moduli space of strongly centred 2-monopoles for even k's and with the moduli space of centred 2-monopoles for odd...

An open manifold M with nonnegative sectional curvature contains a compact totally geodesic submanifold S called the soul. In his solution of the Cheeger-Gromoll conjecture, G. Perelman showed that the metric projection π : M → S was a C Riemannian submersion which coincided with a map previously constructed by V. Sharafutdinov. In this paper we improve Perelman’s result to show that π is in fa...

Main results. Let Tg,n andMg,n denote the Teichmüller and moduli space respectively of genus g Riemann surfaces with n marked points. The Teichmüller metric on these spaces is a natural Finsler metric that quantifies the failure of two different Riemann surfaces to be conformally equivalent. It is equal to the Kobayashi metric [Roy74], and hence reflects the intrinsic complex geometry of these ...

Let F : Σ n × [0, T) → R n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m)...

A submanifold of a Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. The notion of constant-ratio submanifolds was first introduced and studied by the author in [5, 8] during the early 2000s. Such submanifolds relate to a problem in physics concerning the motion in a central force field...