Minimal Submanifolds in Locally Symmetric Spaces

In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.

The Morse index of minimal products of minimal submanifolds in spheres

Tang and Zhang(2020)and Choe and Hoppe(2018)showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds.In this paper,we show that the minimal product is immersed by its first eigenfunctions(of its Laplacian)if and only if the two beginning minimal submanifolds are immersed by their first eigenfunctions.Moreover,we give the estimates of the Morse index and the nullity of the minimal product.In particular,we show that the Clifford minimal submanifold(√n1/nS^(n1).....,√nk/nS^(nk)■S^(n+k-1))has the index(k-1)(n+k+1)and the nullity(k-1)∑_(1≤i<j≤k)(n_(i)+1)(nj+1)(with n=∑n_(j)).

RIGIDITY OF COMPACT MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC AND CONFORMALLY FLAT RIEMANN MANIFOLD

The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].

Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.

Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra

Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O  to be a minimal submanifold of M 6.

On Submanifolds in Locally Symmetric and Conformally Flat Riemannian Manifolds

Let N n+p be an (n+p)-dimensional locally symmetric and conformally flat Riemannian manifold and Mn be an n-dimensional compact submanifold minimally immersed in N n+p . Instead of (n+p)-dimensional unit sphere, we generalize Pinching Theorems about submanifolds in unit sphere and get theorems about submanifolds in locally symmetric and conformally flat Riemannian manifold.

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.

Spectral Characterizations of Veronese Surface in S^4

In this paper, we prove that the Veronese surface can be determined by the 1 spectrum of the Laplace operator.

Equivariant Lagrangian Minimal S^3 in CP^3

The equivariant Lagrangian minimal immersion of Sa into C.Pa is studied. The complete classification and analytic expression for such kinds of immersions are given.