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Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz Ramos) Departament of Geometry and Topology University of Santiago de Compostela Workshop on Hypersurface Geometry and Integrable Systems Sendai 2010

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Page 1: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Real hypersurfaces with constant principal curvatures

in complex projective and hyperbolic spaces

Miguel Domínguez Vázquez (joint work with J. C. Díaz Ramos)

Departament of Geometry and TopologyUniversity of Santiago de Compostela

Workshop on Hypersurface Geometry and Integrable SystemsSendai 2010

Page 2: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Contents

Page 3: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Introduction Homogeneous hypersurfaces vs. hypersurfaceswith constant principal curvatures

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Page 4: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Introduction The problem in space forms

Page 5: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Introduction The problem in complex space forms

Page 6: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

1 2 3 4 5 > 5

1

2

3

4

5

> 5

?

1 2 3 4 5 > 5

1

2

3

4

5

> 5

Introduction

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The problem in complex space forms

Page 7: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

as a symmetric space Root space and Iwasawa decompositions

Page 8: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

as a symmetric space The submanifolds W2n-k

Page 9: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Main Theorem

Page 10: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Main Theorem Idea of the proof

Page 11: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Main Theorem

Page 12: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Contents

Page 13: Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces Miguel Domínguez Vázquez (joint work with J. C. Díaz

Main Theorem Idea of the proof