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 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 The index conjecture for symmetric spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Spaces ◽  
Systematic Approach ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Geodesic Submanifold ◽  
Totally Geodesic Submanifolds ◽  
Higher Rank ◽  
Differential Geom

AbstractIn 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank {\leq 2}, but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional symmetric spaces, Rev. Mat. Iberoam., to appear] we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture, formulated first in [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], for how to calculate the index. The purpose of this paper is to verify the conjecture.

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 Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

2020 ◽  
Author(s):  
Leandro Lichtenfelz ◽  
Gerard Misiołek ◽  
Stephen C Preston
Keyword(s):  
Euler Equations ◽  
Riemannian Geometry ◽  
Exponential Map ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Ideal Fluids ◽  
Volume Preserving ◽  
The Way

Abstract We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

Minimal immersions of S2 and ℝP2 into ℂPn with few higher order singularities

1992 ◽  
Vol 111 (1) ◽  
pp. 93-101 ◽  
Author(s):  
J. Bolton ◽  
L. M. Woodward ◽  
L. Vrancken
Keyword(s):  
Sectional Curvature ◽  
Constant Curvature ◽  
Unit Sphere ◽  
Conformal Structure ◽  
Holomorphic Sectional Curvature ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Minimal Immersions ◽  
Complex Projective

In this paper we extend ideas developed in 2, 4 to study certain minimal immersions of S2 and ℝP2 into ℂPn. Here S2 denotes the unit sphere in ℝ3 with its standard conformal structure and ℝP2 is S2 factored out by the antipodal map, while ℂPn denotes complex projective n-space equipped with the FubiniStudy metric of constant holomorphic sectional curvature 4. Since ℝPn with its standard metric of constant curvature 1 is included in ℂPn as a totally geodesic submanifold, this includes the case of minimal immersions into the unit sphere Sn(1) with its standard metric.

scholarly journals WARPED PRODUCT SEMI-SLANT SUBMANIFOLDS IN LOCALLY RIEMANNIAN PRODUCT MANIFOLDS

2008 ◽  
Vol 77 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MEHMET ATÇEKEN
Keyword(s):  
Warped Product ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Slant Submanifolds ◽  
Geodesic Submanifold

AbstractIn this paper, we prove that there are no warped product proper semi-slant submanifolds such that the spheric submanifold of a warped product is a proper slant. But we show by means of examples the existence of warped product semi-slant submanifolds such that the totally geodesic submanifold of a warped product is a proper slant submanifold in locally Riemannian product manifolds.

scholarly journals On totally geodesic submanifolds in the Jacobian locus

2015 ◽  
Vol 26 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Alessandro Ghigi
Keyword(s):  
Upper Bound ◽  
Positive Characteristic ◽  
Second Fundamental Form ◽  
Shimura Varieties ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
The Second Fundamental Form ◽  
Totally Geodesic Submanifolds ◽  
Cyclic Covers

We study submanifolds of Ag that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map Mg ↪ Ag due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] ∈ Mg in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of Ag obtained from cyclic covers of ℙ1. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.

Infinitesimal rigidity of boundary lattice actions

1999 ◽  
Vol 19 (1) ◽  
pp. 35-60 ◽  
Author(s):  
A. KONONENKO
Keyword(s):  
Symmetric Spaces ◽  
Compact Type ◽  
Infinitesimal Rigidity ◽  
Duality Method ◽  
Cocompact Lattice ◽  
Higher Rank ◽  
Cocompact Lattices ◽  
Lattice Actions ◽  
Cohomological Rigidity ◽  
Twisted Cocycles

In Part 1 we describe a duality method for calculating twisted cocycles. In Part 2 we use our method to prove various results on cohomological rigidity of higher-rank cocompact lattice actions. In Part 3 we use the results of Parts 1 and 2 to prove infinitesimal rigidity of the actions of cocompact lattices on the maximal boundaries of some non-compact type symmetric spaces.

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