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).     

Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds. Mathematics Subject Classification (2000): 53C20, 53C22, 53C24 Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003     

Totally geodesic boundaries of knot complements

Given a compact orientable -manifold whose boundary is a hyperbolic surface and a simple closed curve in its boundary, every knot in is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of is as small as you like.

     

Totally geodesic foliations with nearly-integrable orthogonal distribution

Krasnoyarsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 199–203, January–February, 1991.     

纤维丛的全测地子流形

本文证明了底空间Ｍ是纤维丛Ｐ的全测地子流形；并且在ｄｉｍＰ－ｄｉｍＭ＝２时证明了若Ｐ是平坦的，则Ｐ的每一纤维也是全测地子流形．     

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 geodesic discs in strongly convex domains

We prove that every isometry from the unit disk Δ in ${\mathbb{C}}$ , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ to a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^m}$ , is either holomorphic or anti-holomorphic.     

Totally geodesic foliations close to Riemannian foliations

The relation of the curvature and topology of totally geodesic foliations close to Riemannian ones is studied. The main result complements Ferus's famous theorem on totally geodesic foliations.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 114–118, 1992.     

Totally geodesic Riemannian foliations with locally symmetric leaves

We prove the arithmeticity of totally geodesic Riemannian foliations, with a dense leaf, on complete finite volume Riemannian manifolds when the leaves are isometrically covered by an irreducible symmetric space of noncompact type and rank at least 2. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Totally geodesic hypersurfaces in manifolds of nonpositive curvature

In this paper we determine the structure of an embedded totally geodesic hypersurfaceF or, more generally, of a totally geodesic hypersurfaceF without selfintersections under arbitrarily small angles in a compact manifoldM of nonpositive sectional curvature. Roughly speaking, in the case of locally irreducibleM the result says thatF has only finitely many ends, and each end splits isometrically asK×(0, ∞), whereK is compact. This article was processed by the author using the Springer-Verlag T_EX PJour1g macro package 1991.     

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.     

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.     

On holomorphic families of subspaces of a banach space

In this paper we study holomorphic families of subspaces of a Banach space, which are parametrized by some analytic space. We consider the question of the existence of a holomorphic complement for a given family. It turns out that such a complement does always exist if the basic space is a Stein space, provided for each fixed value of the parameter the subspaces of the family are complemented. From this follows, in particular, the existence of a holomorphic left inverse of a holomorphic operator function, provided such a left inverse exists for each fixed value of the parameter. The latter result gives a positive answer to a question formulated by I.C. Gohberg in a personal conversation with the author. Note of the Editor: This paper was originally published in "Mat. Issled. (Kishinev) 5, vyp 4 (18) (1970), 153–165. The editor is grateful to K. Clancey, W. Kaballo and G. Ph. A. Thijsse for preparing the translation from Russian. Note of the Editor: After the appearance of the original version of the above paper the author informed the editor of Mat. Issled (see Mat. Issled. (Kishinev) 6, vyp (19) (1971), 180) that Corollaries 2 and 3 of Theorem 2 are contained in the papers of G.R. Allan, J. London Math. Soc. 42 (1967), 463–470 and 509–513.     

Totally geodesic embeddings of 7-manifolds in positively curved 13-manifolds

We demonstrate a correspondence between the set of Bazaikin spaces with a subset of Aloff-Wallach and Eschenburg spaces. We show that each Bazaikin space contains at least one and generically 10 totally geodesically embedded Aloff-Wallach or Eschenburg spaces. We use these embeddings to get a sharp upper bound for the pinching of the standard biquotient metrics over the whole family of Bazaikin spaces, and to characterise the curvature properties and regularity of the Bazaikin space in terms of the curvature and regularity of the embedded spaces.Mathematics Subject Classification (2000): 53C20, 53C30, 53C40Acknowledgements. It is a pleasure for us to thank I. Taimanov and W. Ziller for valuable hints and discussion.Dedicated to Ernst Heintze on occasion of his 60th birthday     

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.     

Totally geodesic Kähler immersions in view of curves of order two

In this paper, we characterize of totally geodesic Kähler immersions by extrinsic shapes of some curves having points of order 2.