Totally geodesic submanifolds of maximal rank in symmetric spaces

We gave a complete list of totally geodesic submanifolds of maximal rank in symmetric spaces of noncompact type. The compact cases can be obtained by the duality.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Totally geodesic submanifolds in compact Riemannian symmetric spaces and their stability

In this paper, we determine a large class of totally geodesic submanifolds of a compact Riemannian symmetric space. The stability of these submanifolds in their ambient space is also determined.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Minimality of totally geodesic submanifolds in Finsler geometry

Using the symplectic definition of the Holmes-Thompson volume we prove that totally geodesic submanifolds of a Finsler manifold are minimal for this volume. Thanks to well suited technics the minimality of totally geodesic hypersurfaces (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006) and 2-dimensional totally geodesic surfaces (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006, Ivanov in Algebra i Analiz 13(1)26–38, 2001) had already been proved. However the corresponding statement for the Hausdorff measure is known to be wrong even in the simplest case of totally geodesic 2-dimensional surfaces in a 3-dimensional Finsler manifold (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006).     

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if and are connected locally symmetric spaces of noncompact type where has finite volume, and is a totally geodesic immersion, then the closure of in is an immersed ``algebraic' submanifold. It is also shown that if in addition, the real ranks of and are equal, then the the closure of in is a totally geodesic submanifold of The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.

     

Flat totally geodesic submanifolds of quasisymmetric Siegel domains

Partially supported by an NSF grant.     

Finite energy and totally geodesic maps from locally symmetric spaces of finite volume

LetM=G/K be a locally symmetric space of finite volume and rank 2. We show that any map fromM of weighted finite energy in the sense of Saper can be deformed into a finite energy map. As a consequence such maps can be deformed into totally geodesic ones, and a geometric generalization of Margulis' superrigidity theorem is obtained.     

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.     

The characterization of totally geodesic submanifolds ofS m andCP m

We prove a theorem on ruled surfaces that generalizes a theorem of Ferus on totally geodesic foliations. On the basis of this theorem we obtain criteria for totally geodesic submanifolds ofS ^m andCP ^m that generalize and complement certain results of Borisenko, Ferus, and Abe. We give an application to the geodesic differential forms defined by Dombrowski in the case of submanifolds ofS ^m andCP ^m.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 106–116.The author is grateful to V. A. Toponogov for posing this problem and for attention to the work and to A. A. Borisenko for helpful criticisms.     

Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPⁿ, compact minimal quaternionic coisotropic submanifolds in ℚPⁿ and compact minimal hypersurfaces in P² (Cay).      

n-dimensional totally real minimal submanifolds of CP n

Let CP ⁿ be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CP ⁿ . In this paper we prove the following results.

Almost geodesic mappings of Riemann spaces onto symmetric Riemann spaces

It is shown that if a Riemann space V_n admits a reduced almost geodesic mapping Π₂ onto a symmetric Riemann space ¯V_n, then ¯V_n has constant curvature, and V_n is itself a symmetric space.