scholarly journals Helgason spheres of compact symmetric spaces and immersions of finite type

2001 ◽  
Vol 63 (2) ◽  
pp. 243-255
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Finite Type ◽  
Symmetric Spaces ◽  
Unit Vector ◽  
Geodesic Sphere ◽  
Compact Type ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Unit Speed

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

scholarly journals Reflections and symmetries in compact symmetric spaces

1988 ◽  
Vol 38 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Bang-Yen Chen ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces

Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

scholarly journals On splitting rank of non-compact-type symmetric spaces and bounded cohomology

2018 ◽  
Vol 12 (02) ◽  
pp. 465-489
Author(s):  
Shi Wang
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Maximal Dimension ◽  
Bounded Cohomology ◽  
Compact Type ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Higher Rank

Let [Formula: see text] be a higher rank symmetric space of non-compact type where [Formula: see text]. We define the splitting rank of [Formula: see text], denoted by [Formula: see text], to be the maximal dimension of a totally geodesic submanifold [Formula: see text] which splits off an isometric [Formula: see text]-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map [Formula: see text] is surjective in degrees [Formula: see text], provided [Formula: see text] has no direct factors of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

scholarly journals Homotopy of compact symmetric spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 221-228 ◽  
Author(s):  
John M. Burns
Keyword(s):  
Symmetric Spaces ◽  
Unified Approach ◽  
Topological Characterization ◽  
New Approach ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Compact Symmetric Spaces

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

scholarly journals On the index of symmetric spaces

2018 ◽  
Vol 2018 (737) ◽  
pp. 33-48 ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Riemannian Symmetric Spaces

AbstractLetMbe an irreducible Riemannian symmetric space. The index ofMis the minimal codimension of a (nontrivial) totally geodesic submanifold ofM. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.

scholarly journals CONVOLUTIONS OF GENERIC ORBITAL MEASURES IN COMPACT SYMMETRIC SPACES

2009 ◽  
Vol 79 (3) ◽  
pp. 513-522 ◽  
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Invariant Measure ◽  
Symmetric Space ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Nonempty Interior ◽  
Absolutely Continuous ◽  
Double Cosets ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces ◽  
Dense Set

AbstractWe prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

scholarly journals Some Topological and Algebraic Features of Symmetric Spaces

2020 ◽  
pp. 144-155
Author(s):  
Um Salama ◽  
Ahmed Abd Alla ◽  
A. Elemam
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Lie Algebras ◽  
Symmetric Spaces ◽  
Algebraic Approach ◽  
Root Systems ◽  
Algebraic Properties ◽  
Important Field

In this study, we introduce some approaches, geometrical and algebraic, which help to give further understanding of symmetric spaces. Symmetric space is a very important field for understanding abstract and applied features of spaces. We have introduced Riemannian Manifold, Lie groups and Lie algebras, and some of their topological and algebraic properties, with some concentration on Lie algebras and root systems , which help classification and many applications of symmetric spaces. The paper is an attempt to explain some algebraic features of symmetric spaces and how to get some of their properties using algebraic approach, concluded with some results.

scholarly journals Symmetric Spaces in Riemannian and Semi-Riemannian Geometry

2020 ◽  
pp. 47-59
Author(s):  
Ehsan Hashempour ◽  
Mir Mohammad Seyedvalilo
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Riemannian Geometry ◽  
Symmetric Spaces ◽  
Sufficient Conditions ◽  
Local Symmetry ◽  
Symmetric Pair ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

In this paper, we will obtain the necessary and sufficient conditions for the analysis of the position of local symmetry on an arbitrary Riemannian manifold. These conditions are devoid of the aspects of Lie groups, and thus can be used in calculations of procedures, without interfering with the concepts of Lie groups, and improve intuitive attitudes. Also, we will study and create equivalent conditions for a situation where a two-metric homogeneous Riemannian manifold is located symmetrically. In addition, in this paper it is stated that the symmetric space (M, g) can be seen as a homogeneous space G/K. Also, one-to-one correspondence between the symmetric space and the symmetric pair is shown, and curvature is studied on a symmetric space.

scholarly journals Dynamics of the heat semigroup on symmetric spaces

2009 ◽  
Vol 30 (2) ◽  
pp. 457-468 ◽  
Author(s):  
LIZHEN JI ◽  
ANDREAS WEBER
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Chaotic Behavior ◽  
Heat Semigroup ◽  
Euclidean Spaces ◽  
Compact Type ◽  
Heat Semigroups

AbstractThe aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

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