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.

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 On the dimension of totally geodesic submanifolds in the Prym loci

2021 ◽  
Author(s):  
Elisabetta Colombo ◽  
Paola Frediani
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Double Cover ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Geodesic Submanifold ◽  
Totally Geodesic Submanifolds ◽  
Double Covers

AbstractIn this paper we give a bound on the dimension of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. This improves the already known bounds. The idea is to adapt the techniques introduced by the authors in collaboration with A. Ghigi and G. P. Pirola for the Torelli map to the case of the Prym maps of (ramified) double covers.

scholarly journals Totally geodesic submanifolds in tangent bundle with g-natural metric

2014 ◽  
Vol 11 (09) ◽  
pp. 1460033 ◽  
Author(s):  
Stanisław Ewert-Krzemieniewski
Keyword(s):  
Vector Field ◽  
Tangent Bundle ◽  
Sufficient Condition ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Base Manifold ◽  
Geodesic Submanifolds ◽  
Geodesic Submanifold ◽  
Totally Geodesic Submanifolds

In a tangent bundle endowed with g-natural metric G we investigate submanifolds defined by a vector field given on a base manifold. We give a sufficient condition for a vector field on M to define totally geodesic submanifold in (TM, G). The parallel vector field is discussed in more detail.

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.

THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

2007 ◽  
Vol 09 (02) ◽  
pp. 183-200 ◽  
Author(s):  
YOUNG JIN SUH ◽  
HAE YOUNG YANG
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Minimal Hypersurface ◽  
Second Fundamental Form ◽  
Geodesic Sphere ◽  
Clifford Torus ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces ◽  
The Second Fundamental Form

In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

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.

On non-invariant hypersurfaces of a nearly hyperbolic Sasakian manifold

2017 ◽  
Vol 28 (08) ◽  
pp. 1750064
Author(s):  
Mobin Ahmad ◽  
Shadab Ahmad Khan ◽  
Toukeer Khan
Keyword(s):  
Fundamental Form ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Second Fundamental Form ◽  
Text Structure ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Necessary And Sufficient

We consider a nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with [Formula: see text]-structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with [Formula: see text]-structure under the condition when f is parallel.

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.

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

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