Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.     

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.     

Extrinsic homogeneity of parallel submanifolds II

We discuss the question whether a (complete) parallel submanifold M of a Riemannian symmetric space N is an (extrinsically) homogeneous submanifold, i.e. whether there exists a subgroup of the isometries of N which acts transitively on M. In a previous paper, we have discussed this question in case the universal covering space of M is irreducible. It is the subject of this paper to generalize this result to the case when the universal covering space of M has no Euclidian factor.     

A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.

     

The geometry of homogeneous submanifolds of hyperbolic space

We prove, in a purely geometric way, that there are no connected irreducible proper subgroups of SO(N,1). Moreover, a weakly irreducible subgroup of SO(N,1) must either act transitively irreducible subgroup of SO(N,1) must either act transitively on the hyperbolic space or on a horosphere. This has obvious consequences for Lorentzian holonomy and in particular explains clasification results of Marcel Berger's list (e.g. the fact that an irreducible Lorentzian locally symmetric space has constant curvatures). We also prove that a minimal homogeneous submanifold of hyperbolic space must be totally-geodesic. Received August 10, 1999; in final form November 23, 1999 / Published online March 12, 2001     

The Kenmotsu hypersurfaces axiom for 6-dimensional Hermitian submanifolds of the Cayley algebra

We prove that every 6-dimensional Hermitian submanifold of the Cayley algebra satisfying the Kenmotsu Hypersurfaces Axiom is a locally symmetric submanifold of Ricci type.     

Some results on the occurrence of compact minimal submanifolds

Varying the situation considered in Myers theorem, we show, via standard index form techniques, that a complete Riemannian manifold which admits a compact minimal submanifold is necessarily compact, provided a suitable curvature object is positive on the average along the geodesies issuing orthogonally from the minimal submanifold. By slightly recasting this result, one establishes the nonexistence of compact minimal submanifolds (in particular, closed geodesies) in complete noncompact manifolds which obey an appropriate curvature condition. A generalization of a result of Tipler concerning the occurrence of zeros of solutions to the scalar Jacobi equation is also obtained.     

Concentration on minimal submanifolds for a Yamabe-type problem

We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.     

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of . We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.     

Almost invariant submanifolds for compact group actions

We define a C ¹ distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C ¹-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros of sections of extended normal bundles. Received September 14, 1999 / final version received November 29, 1999     

Isotropic submanifolds with pointwise planar normal sections

Submanifolds of E^m with pointwise planar normal sections were studied in [1] and others. In the present paper, we will prove that an isotropic submanifold in E^m with pointwise planar normal sections is isometric to a symmetric space of rank one or to a Euclidean space. Moreover we will determine such surfaces in E^m with the above assumptions.     

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P ^m in ambient Riemannian spaces N ⁿ with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.     

Twisted dirac operator on minimal submanifolds

Hijazi and Zhang improved Friedrich’s inequality for non-minimal spin submanifolds. Their proof relies on the non-minimality assumption. We use another method to prove that their theorem holds also for minimal submanifolds. As an application, we show that any Kähler manifold can be embedded as a totally geodesic submanifold of its twistor space and apply the above result.     

Maximum principles and nonexistence results for minimal submanifolds

We construct a quadratic form on ℝ^n+k of signature (n-k) which is subharmonic on any n-dimensional minimal submanifold in ℝ^n+k. This yields an improvement over the convex hull property of minimal submanifolds as well as necessary conditions for compact minimal submanifolds the boundaries of which lie in disconnected sets. The argument also extends to submanifolds of bounded mean curvature. Furthermore an optimal nonexistence result is derived by employing a different geometrical argument, which is based on the construction of n-dimensional catenoids.     

Spectrum comparison on free boundary minimal submanifolds of Euclidean domains

In this paper, using ideas of Simons, Ros, and Savo, we prove a comparison between the spectrums of the stability operator and the Hodge–Laplacian acting on differential 1-forms on a compact minimal submanifold immersed into a Euclidean domain.     

The submanifolds of compact operators with fixed Jordan cells

The manifold of symmetric real matrices with fixed multiplicities of eigenvalues was first considered by V. I. Arnold. In the case of compact real self-adjoint operators, his results were generalized by the group of Japanese mathematicians, D. Fujiwara, M. Tanikawa, and Sh. Yukita. They introduced a special local diffeomorphism that maps Arnold’s submanifold to a flat subspace. Ya. Dymarskii developed the aforementioned works into a full theory. Here, we will describe the smooth structure of a submanifold of the compact operators of the general form such that the selected eigenvalue corresponds to a fixed Jordan normal form. The research is based on a straightening diffeomorphism and Arnold’s results about families of matrices depending on parameters.     

On mean curvatures in submanifolds geometry

By using moving frame theory,first we introduce 2p-th mean curvatures and(2p 1)-th mean curvature vector fields for a submanifold.We then give an integral expression of them that characterizes them as mean values of symmetric functions of principle curvatures.Next we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.     

On the curvature of minimal submanifolds in a sphere

S. T. Yau proved inAmer. J. Math. 97 (1975), p. 95, Theorem 15 that if the sectional curvature of ann-dimensional compact minimal submanifold in the (n + p)-dimensional unit sphere is everywhere greater than (p – 1)/(2p – 1), then this minimal submanifold is totally geodesic. In this note we improve this bound for the casep 2 to (3p – 2)/(6p).     

Complete space-like submanifolds in locally symmetric semi-definite spaces

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized     

Extrinsic homogeneity of parallel submanifolds

We consider parallel submanifolds M of a Riemannian symmetric space N and study the question whether M is extrinsically homogeneous in N, i.e. whether there exists a subgroup of the isometry group of N which acts transitively on M. Provided that N is of compact or non-compact type, we establish the extrinsic homogeneity of every complete irreducible parallel submanifold of N whose dimension is at least three and which is not contained in any flat of N.