Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Totally umbilical manifolds inG(2,n). I

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds F^l of dimension l 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H 0, H 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S¹ × S¹ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic S^l+1 × S^l+1 G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold F^l E^l+2 (l 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 83–98, 1991.     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001     

Submanifolds with restrictions on extrinsic <Emphasis Type="Italic">q</Emphasis>th scalar curvature

We study the structure of the minimum set of the normal curvature for a symmetric bilinear map on Euclidean or Hilbert space, the conditions when this set contains strongly umbilical, conformal nullity, etc. linear subspaces. The main goals are estimates from above of the codimension of these subspaces for a symmetric bilinear map with positive normal curvature and the inequality type restriction on the extrinsic qth scalar curvature. We estimate from above the codimension of asymptotic and relative nullity subspaces for a symmetric bilinear map with nonpositive extrinsic qth scalar curvature. Applying the algebraic results to the second fundamental form of a submanifold with low codimension, we characterize the totally umbilical and totally geodesic submanifolds, prove local nonembedding theorems for the products of Riemannian manifolds and global extremal theorem for the space of positive curvature. On the way we generalize results by Florit (1994), Borisenko (1977, 1987) and Okrut (1991) about Riemannian and Hilbert submanifolds. The research was supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel; and by Center for Computational Mathematics and Scientific Computation, University of Haifa.     

Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces. Authors’ addresses: Mirjana Djorić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade, Serbia; Masafumi Okumura, 5-25-25 Minami Ikuta, Tama-ku, Kawasaki, Japan     

Deformation of a submanifold in a Euclidean space with fixed gauss image: II

Deformation of an m-dimensional submanifold immersed in a Euclidean n-space is studied when the Gauss image does not move pointwise. The present paper is the third one and contains deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformations of totally-real submanifolds in a complex space.     

Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele

1. We show that symmetric R-spaces can. be imbedded in euclidean space with parallel second fundamental tensor (Dα=0).
2. We give a restrictive necessary condition for totally geodesic submanifolds of the Grassmannian to be the Gauss image of an immersion with Dα=0, c.f. [9].

     

Induced generalized <Emphasis Type="Italic">S</Emphasis>-space-form structures on submanifolds

We study non-anti-invariant slant submanifolds of generalized S-space-forms with two structure vector felds in order to know if they inherit the ambient structure. In this context, we focus on totally geodesic, totally umbilical, totally ƒ-geodesic and totally ƒ-umbilical non-anti-invariant slant submanifolds and obtain some obstructions. Moreover, we present some new interesting examples of generalized S-space-forms.     

Reconstruction from Line Integrals in Spaces of Constant Curvature

New reconstruction formula for the line integral transformation in Euclidean spaces is found. The general k-plane integral transform in Euclidean space is related to a totally geodesic integral transform for an arbitrary Riemannian space of constant curvature by means of a factorization property. Duality theorems for the totally geodesic transforms are stated.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R³.     

On Hamiltonian stable minimal Lagrangian surfaces in CP2

We study Hamiltonian stable minimal Lagrangian closed submanifolds in the standard complex projective n-space CP ⁿ .It is shown that when n = 2such a surface Σis either totally geodesic or flat if the multiplicity of the Laplacian acting on C∞(Σ)is less than or equal to 6.     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002     

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.     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

On the canonical extension of a differentiable manifold

In this paper, the differential geometry of second canonical extension² M of a differentiable manifoldM is studied. Some vector fields tangent to² M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and² M respectively.     

On spectral geometry of minimal parallel submanifolds

We consider the problem of characterizing some minimal submanifolds using the spectrum⁰Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCP ⁿ, 3≤n≤6, by their⁰Spec in the class of all compact totally real minimal submanifolds ofCP ⁿ. Moreover, we characterize the Clifford torus by its⁰Spec in the class of all compact minimal submanifolds of the Euclidean sphereS ⁿ⁺¹(1). Authors supported by funds of the University of Lecce and the M.U.R.S.T.