Differential Geometry 1st edition by Alvarez Lopez Jesus A ISBN 9814261173 9789814261173

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ISBN 10: 9814261173

ISBN 13:9789814261173

Author:Alvarez Lopez Jesus A

This volume contains research and expository papers on recent advances in foliations and Riemannian geometry. Some of the topics covered in this volume include: topology, geometry, dynamics and analysis of foliations, curvature, submanifold theory, Lie groups and harmonic maps.Among the contributions, readers may find an extensive survey on characteristic classes of Riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of Thurston’s and Thurston-Bennequin’s inequalities, a discussion about Fatou-Julia decompositions for foliations and a description of singular Riemannian foliations on spaces without conjugate points.Papers on submanifold theory focus on the existence of graphs with prescribed mean curvature and mean curvature flow for spacelike graphs, isometric and conformal deformations and detailed surveys on totally geodesic submanifolds in symmetric spaces, cohomogeneity one actions on hyperbolic spaces and rigidity of geodesic spheres in space forms. Geometric realizability of curvature tensors and curvature operators are also treated in this volume with special attention to the affine and the pseudo-Riemannian settings. Also, some contributions on biharmonic maps and submanifolds enrich the scope of this volume in providing an overview of different topics of current interest in differential geometry.

Differential Geometry 1st Table of contents:

1. Introduction

2. Classifying spaces

3. Primary classes

4. Secondary classes

5. Variation of secondary classes

6. Molino Structure Theory

7. Some open problems

References

Non unique-ergodicity of harmonic measures: Smoothing Samuel Petite’s examples

B. Deroin

1. Introduction

2. The construction

3. Questions

Acknowledgement

References

On the uniform simplicity of diffeomorphism groups T. Tsuboi

1. Introduction

2. Uniform perfectness of diffeomorphism groups

3. Uniform simplicity of the diffeomorphism groups

Acknowledgement

References

On Bennequin’s isotopy lemma and Thurston’s inequality Y. Mitsumatsu

1. Thurston’s Inequality

2. Thurston’s Absolute Inequality for Spinnable Foliations

3. Dehn Filling

4. Bennequin’s Isotopy Lemma

Acknowledgements

References

On the Julia sets of complex codimension-one transversally holomorphic foliations T. Asuke

1. Introduction

2. Definition of the Fatou and Julia sets

3. Some properties of Julia sets

4. Examples

References

Singular Riemannian foliations on spaces without conjugate points A. Lytchak

1. Introduction

2. Preliminaries

3. Geometric arguments

4. Topological arguments

Acknowledgments

References

Variational formulae for the total mean curvatures of a codimension-one distribution V. Rovenski and

0. Introduction

1. Main Results

2. Proofs

2.1. Algebraic preliminaries

2.2. Proof of Theorem 0.2

2.3. Variational formulae

References

On a Weitzenböck-like formula for Riemannian foliations V. Slesar

1. Introduction

2. Canonical differential operators defined on a Riemannian foliation

3. Adiabatic limits and Riemannian foliations

4. A transversal Weitzenböck formula

Acknowledgments

References

Duality and minimality for Riemannian foliations on open manifolds X. M. Masa

1. The duality

2. The minimality

3. Singular Riemannian foliations

References

Open problems on foliations

Part B Riemannian geometry

Graphs with prescribed mean curvature M. Dajczer

1. Introduction

2. Killing graphs

3. Riemannian submersions

4. Conformal Killing graphs

References

Genuine isometric and conformal deformations of submanifolds R. Tojeiro

1. Introduction

2. The hypersurface case

2.1. Local theory for the isometric case

2.2. Global theory of isometrically deformable hypersurfaces

2.3. Digression: Extending intrinsic isometries

2.4. Local theory of conformal deformations of hypersurfaces

3. Higher codimensions: Rigidity results

3.1. The s-nullity and the conformal s-nullity

3.2. A main tool: flat bilinear forms

3.3. Conformal geometry in the light cone

4. The general deformation problem

5. Genuine deformations

5.1. Ruled extensions

5.2. Genuine conformal deformations of submanifolds

5.3. Constructing conformal pairs from isometric ones

Acknowledgment

References

Totally geodesic submanifolds in Riemannian symmetric spaces S. Klein

1. Totally geodesic submanifolds

2. Maximal totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2

2.1. G +2 (Rn+2)

2.2. G2 (Cn+2)

2.3. G (Hn+2)

2.4. SU(3)=SO(3)

2.5. SU(6)=Sp(3)

2.6. SO(10)=U(5)

2.7. E6=(U(1) Spin(10))

2.8. E6=F4

2.9. G2=SO(4)

2.10. SU(3)

2.11. Sp(2)

2.12. G2

Acknowledgments

References

The orbits of cohomogeneity one actions on complex hyperbolic spaces J. C. Díaz-Ramos

1. The geometry of the orbits

2. Hypersurfaces with constant principal curvatures

Acknowledgements

References

Rigidity results for geodesic spheres in space forms J. Roth

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.1

4. Rigidity results in the Euclidean space

Acknowledgements

References

Mean curvature flow and Bernstein-Calabi results for spacelike graphs G. Li and I. M. C. Salavessa

1. Introduction

2. Bernstein-Calabi and Heinz-Chern type results

3. The mean curvature flow

4. Homotopy to a constant map

Acknowledgements

References

Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators P.

1. Introduction

1.1. Realizing Riemannian algebraic curvature tensors

1.2. Osserman geometry

1.3. Affine geometry

1.4. Torsion free connections and Riemannian geometry

2. The proof of Theorem 1.3

Acknowledgments

References

Conformally Osserman multiply warped product structures in the Riemannian setting M. Brozos-Vázquez

1. Introduction

2. Preliminaries

3. Conformally Osserman multiply warped products

4. Locally conformally at multiply warped products

5. Osserman multiply warped products

6. Multiply warped products of constant curvature

Acknowledgements

References

Riemannian -symmetric spaces M. Goze and E. Remm

1. Riemannian reductive homogeneous spaces

2. Riemannian -symmetric spaces

2.1. -symmetric spaces

2.2. -grading of the Lie algebra g of G

2.3. Riemannian and Indefinite Riemannian -symmetric spaces

2.4. Irreducible Riemannian -symmetric spaces

3. Classification of compact simple Z2 symmetric spaces

4. On the classification of Riemannian compact Z2 -symmetric spaces

4.1. Z2 -symmetric metrics on flag manifolds

4.2. The Z2 -Riemannian symmetric space SO(2m)=Sp(m)

References

Methods for solving the Jacobi equation. Constant osculating rank vs. constant Jacobi osculating ran

1. Introduction and preliminaries

2. Preliminaries about H-type groups

3. Kaplan’s example

3.1. Constant osculating rank of the Jacobi operator along a special family of geodesics

3.2. Resolution of the Jacobi equation

3.3. Relation between both methods

Acknowledgements

References

On the reparametrization of affine homogeneous geodesics Z. Dusek

1. Introduction

2. Homogeneous geodesics in pseudo-Riemannian manifolds

3. Homogeneous geodesics in affine manifolds

4. Locally homogeneous connections in dimension two

5. G.o. manifolds of type A

6. G.o. manifolds of type B

7. General connection of type B

Acknowledgments

References

Conjugate connections and differential equations on infinite dimensional manifolds M. Aghasi, C. T.

1. Introduction

2. Preliminaries

3. Classification for vector bundle structures of T2M

4. Connections and ordinary differential equations

5. The Earle and Eells foliation theorem in Fréchet spaces

References

Totally biharmonic submanifolds D. Impera and S. Montaldo

1. Introduction

2. Biharmonic maps

3. Totally biharmonic hypersurfaces

4. Totally biharmonic surfaces of space forms

5. Biharmonic curves in H3

Acknowledgements

References

The biharmonicity of unit vector fields on the Poincaré halfspace Hn M. K. Markellos

1. Introduction

2. Preliminaries

2.1. Biharmonic maps

2.2. The tangent bundle and the unit sphere bundle

3. Homogeneous structures

Acknowledgements

References

Perspectives on biharmonic maps and submanifolds A. Balmus

1. Introduction

2. Bihamonic maps and warped product manifolds

3. Biharmonic submanifolds in space forms

4. On the biharmonicity of the Gauss map

Acknowledgements

References

Contact pair structures and associated metrics G. Bande and A. Hadjar

1. Introduction

2. Preliminaries on contact pairs

3. Contact pair structures and almost contact structures

3.1. Almost contact structures

3.2. Contact pair structures

4. Compatible and associated metrics

4.1. Orthogonal foliations

Final comments

Acknowledgements

References

Paraquaternionic manifolds and mixed 3-structures S. Ianus and G. E. Vilcu

1. Introduction

2. Paraquaternionic structures on manifolds

3. Manifolds endowed with mixed 3-structures

4. Normal semi-invariant submanifolds and mixed 3-structures

Acknowledgements

References

On topological obstruction of compact positively Ricci curved manifolds W.-H. Chen

1. Introduction

2. An Extension of Myers’ Theorem

Acknowledgements

References

Gray curvature conditions and the Tanaka-Webster connection R. Mocanu

1. Different types of Gray curvature conditions

2. Gray curvature conditions for the Tanaka-Webster connection

References

Riemannian structures on higher order frame bundles from classical linear connections J. Kurek and W

1. Introduction

2. Natural operators

3. The main result

References

Distributions on the cotangent bundle from torsion-free connections J. Kurek and W. M. Mikulski

1. Introduction

2. The main result

3. Proof of the main result

References

On the geodesics of the rotational surfaces in the Bianchi-Cartan-Vranceanu spaces P. Piu and M. M.

1. Introduction

2. Geodesics on surfaces of revolution

3. The Clairaut’s relation

4. Geodesics on the cylinder

References

Cotangent bundles with general natural Kähler structures of quasi-constant holomorphic sectional cu

1. Introduction

2. The quasi-constant holomorphic sectional curvatures of the cotangent bundles with general natural

Acknowledgements

References

Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space M. I. Munteanu and A. I.

1. Preliminaries

2. Weingarten translation surfaces

Acknowledgements

References

G-structures defined on pseudo-Riemannian manifolds I. Sánchez-Rodríguez

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