In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both CPn and hyperquadric CPn. The moduli space all those noncongruent ones introduced, which can be described by certain symmetric modulo an appropriate group action. Using description, many examples, such as holomorphic higher degree, nonhomogenous constant cu...

Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...

We introduce an evolution equation which deforms metrics on 3-manifolds with sectional curvature of one sign. Given a closed 3-manifold with an initial metric with negative sectional curvature, we conjecture that this flow will exist for all time and converge to a hyperbolic metric after a normalization. We shall establish a monotonicity formula in support of this conjecture. Note that in contr...

Let F : Σ n × [0, T) → R n+m be a family of compact immersed sub-manifolds moving by their mean curvature vector. 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 investigate higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m) p...

Let G be a locally compact group. The notion of "totally real" unitary representation of G is defined and investigated in §1. In particular, if K is a compact subgroup of G, it is shown that every closed G-invariant subspace of L2(G/K) is spanned by real-valued functions if, and only if, KgK=Kg~K for every geG. In §2 the coset space X=G/K is specialized to a Riemannian symmetric space, where th...

We generalize a Bernstein-type result due to Albujer and Alı́as, for maximal surfaces in a curved Lorentzian product 3-manifold of the form Σ1 ×R, to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f : Σ1 → Σ2 between two Riemannian manifolds (Σ1 ,g1) and (Σ n 2,g2) of sectional curvatures K1 and K2, respecti...