# Symmetry and shape

According to Felix Klein, geometry is the study of those properties of a space that are invariant under the action of a group of transformations. In Riemannian geometry the group of transformations of the space is called the isometry group, although the space can sometimes have additional properties arising from mathematical or physical requirements that also need to be preserved by the transformation group. This is the case of a submanifold, whose geometry is influenced by the structure of the ambient space.

Intuitively, symmetry is the correspondence of shape at each point of a space. An important problem in geometry and in several physical sciences is to determine the symmetries of a space in terms of its shape. Submanifolds that are invariant under the isometries of the ambient manifold are called extrinsically homogeneous, or just homogeneous when there is no possibility of confusion. The main idea of my research project is to classify homogeneous submanifolds of a given Riemannian manifold, study the geometric properties of homogeneous submanifolds, and determine whether their shape is characteristic of them.

## Funding

This topic of research has been funded by the following fellowships and projects:

*Simetría, curvatura y ecuaciones diferenciales en geometría (Symmetry, curvature and differential equations in geometry)*, Ministry of Science and Innovation MTM2016-75897-P (Spain) (72963 euro), 30 December 2016 - 29 December 2019. Principal investigator: José Carlos Díaz Ramos, Eduardo García Río.*Simetría, homogeneidad y curvatura (Symmetry, homogeneity and curvature)*, Galician government EM2014/009 (Spain) (93000 euro), 14 May 2014 - 14 May 2017. Principal investigator: José Carlos Díaz Ramos.*Simetría, curvatura y rigidez de estructuras geométricas (Symmetry, curvature and rigidity of geometric structures)*, Ministry of Science and Innovation MTM2013-41335-P (Spain) (47700 euro), 1 January 2014 - 31 December 2016. Principal investigator: José Carlos Díaz Ramos, Eduardo García Río.*Isometric actions*, European Commission PERG04-GA-2008-239162 (45000 euro), 1 April 2009 - 31 March 2012. Principal investigator: José Carlos Díaz Ramos.*Symmetry and shape*, European Commission MEIF-CT-2006-038754 (147640.42 euro), 1 March 2007 - 28 December 2008. Principal investigator: Jürgen Berndt.

## Advisorship of students

I have been advisor of the following Ph.D. students:

*Cristina Vidal Castiñeira*: Submanifolds in complex projective and hyperbolic planes, Ph.D. thesis, University of Santiago de Compostela, .*Mohammad Javad Vanaei*: Cohomogeneity one Lorentzian Manifolds of Constant Curvature, Ph.D. thesis, Tarbiat Modares University, .*Miguel Domínguez Vázquez*: Isoparametric foliations and polar actions on complex space forms, Ph.D. thesis, University of Santiago de Compostela, .

I have also been advisor of the following research works:

*Olga Pérez Barral*: Acciones isométricas en el espacio-tiempo de Minkowski (Isometric actions on Minkowski spacetimes), master thesis, University of Santiago de Compostela, .*Víctor Sanmartín López*: Hipersuperficies isoparamétricas en el espacio hiperbólico complejo (Isoparametric hypersurfaces in the complex hyperbolic space), master thesis, University of Santiago de Compostela, .*Cristina Vidal Castiñeira*: Hipersuperficies reais con dúas curvaturas principais nos planos proxectivo e hiperbólico complexos (Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes), master thesis, University of Santiago de Compostela, .*Miguel Domínguez Vázquez*: Hipersuperficies con curvaturas principais constantes nos espazos proxectivo e hiperbólico complexos (Hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces), DEA, University of Santiago de Compostela, .

## Publications

A list of publications related to this research project is presented below. For a whole list of my publications (with the pdf of some of my papers) click here.

### Research papers

- J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Isoparametric submanifolds in two-dimensional complex space forms, to appear in
*Ann. Glob. Anal. Geom.* - J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes, to appear in
*Ann. Mat. Pura Appl.* - M. Domínguez-Vázquez, C. Gorodski: Polar foliations on quaternionic projective spaces, to appear in
*Tohoku Math. J.* - V. Sanmartín-López: Spacelike isoparametric hypersurfaces, to appear in
*Differential Geom. Appl.* - J. C. Díaz-Ramos, M. Domínguez-Vázquez, A. Kollross: Polar actions on complex hyperbolic spaces,
*Math. Z.***287**(2017), 1183-1213. - J. C. Díaz-Ramos, M. Domínguez-Vázquez, V. Sanmartín-López: Isoparametric hypersurfaces in complex hyperbolic spaces,
*Adv. Math.***314**(2017), 756-805. - J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes,
*J. Geom. Anal.***27**(2017), 442-465. - J. Berndt, J. C. Díaz-Ramos, M. Vanaei: Cohomogeneity one actions on Minkowski spaces,
*Monatsh. Math.***184**(2017), 185-200. - J. C. Díaz-Ramos, S. M. B. Kashani, M. Vanaei: Cohomogeneity one actions on anti de Sitter spacetimes,
*Results Math.***72**(2017), 515-536. - M. Domínguez-Vázquez: Isoparametric foliations on complex projective spaces,
*Trans. Amer. Math. Soc.***368**(2016), 1211-1249. - J. C. Díaz-Ramos, C. Vidal-Castiñeira: Examples of real hypersurfaces with two principal curvatures in complex projective planes,
*Int. J. Geom. Meth. Mod. Physics***12**(2015), 1560013 (6 pages). - M. Domínguez-Vázquez: Canonical extension of submanifolds and foliations in noncompact symmetric spaces,
*Int. Math. Res. Not. (IMRN)***22**(2015), 12114-12125. - J. Berndt, M. Domínguez-Vázquez: Cohomogeneity one actions on some noncompact symmetric spaces of rank two,
*Transform. Groups***20**(2015), 921-938. - J. C. Díaz-Ramos, M. Domínguez-Vázquez: Isoparametric hypersurfaces in Damek-Ricci spaces,
*Adv. Math.***239**(2013), 1-17. - J. Berndt, J. C. Díaz-Ramos: Polar actions on the complex hyperbolic plane,
*Ann. Global Anal. Geom.***43**(2013), 99-106. - J. Berndt, J. C. Díaz-Ramos: Homogeneous polar foliations of complex hyperbolic spaces,
*Comm. Anal. Geom.***20**(2012), 435-454. - J. C. Díaz-Ramos, M. Domínguez-Vázquez: Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces,
*Math. Z.***271**(2012), 1037-1042. - J. C. Díaz-Ramos, M. Domínguez-Vázquez: Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms,
*Indiana Univ. Math. J.***60**(2011), 859-882. - J. C. Díaz-Ramos, A. Kollross: Polar actions with a fixed point,
*Differential Geom. Appl.***29**(2011), 20-25. - M. Domínguez-Vázquez: Real hypersurfaces with constant principal curvatures in complex space forms,
*Differential Geom. Appl.*, 29, suppl. 1, (2011), 65-70. - J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type,
*J. Differential Geom.***86**(2010), 191-235. - J. Berndt, J. C. Díaz-Ramos: Homogeneous hypersurfaces in complex hyperbolic spaces,
*Geom. Dedicata***138**(2009), 129-150. - J. Berndt, J. C. Díaz-Ramos: Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane,
*Proc. Amer. Math. Soc.***135**(2007), 3349-3357. - J. Berndt, J. C. Díaz-Ramos: Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces,
*J. London Math. Soc.***74**(2006), 778-798.

### Conference proceedings

- J. C. Díaz-Ramos, M. Domínguez-Vázquez, V. Sanmartín-López: Anti-De Sitter spacetimes and isoparametric hypersurfaces in complex hyperbolic spaces, to appear in
*Lorentzian Geometry and Related Topics: GeLoMa 2016, Málaga, Spain, September 20-23*. - J. C. Díaz-Ramos: Polar Actions on Symmetric Spaces,
*Recent Trends in Lorentzian Geometry, Springer Proceedings in Mathematics & Statistics*, Springer, 2013, 315-334. - M. Domínguez-Vázquez: On the existence of inhomogeneous isoparametric foliations of higher codimension on complex projective spaces,
*Pure and Applied Differential Geometry - PADGE 2012 (In Memory of Franki Dillen)*, Shaker Verlag, 2013, 114-125. - M. Domínguez-Vázquez: Some interesting families of non-Hopf real hypersurfaces in complex space forms,
*Proceedings of The Seventeenth International Workshop on Differential Geometry and related fields***17**, 2013, 113-124. - M. Domínguez-Vázquez: On isoparametric hypersurfaces in complex hyperbolic spaces,
*International Meeting on Differential Geometry (Córdoba 2010)*, Universidad de Córdoba, 2013, 71-92. - J. C. Díaz-Ramos: Polar actions on complex space forms,
*Proceedings of The Sixteenth International Workshop on Differential Geometry and related fields***16**, 2012, 69-88. - M. Domínguez-Vázquez: On an inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane,
*XX International Fall Workshop on Geometry and Physics, AIP Conference Proceedings***1460**, Amer. Inst. Phys., 2012, 169-173. - J. C. Díaz-Ramos: The orbits of cohomogeneity one actions on complex hyperbolic spaces,
*Proceedings of the VIII International Colloquium on Differential Geometry*, World Sci. Publ. Co., 2009, 146-155.

## Talks

This is a list of talks related to this research project. For a more complete list of my talks click here.

### Conference talks

- J. C. Díaz-Ramos:
*Homogeneous submanifolds of complex space forms*, The third Japanese-Spanish workshop on Differential Geometry, Madrid, Spain, 18-22 September 2017. - M. Domínguez-Vázquez:
*Isoparametric hypersurfaces in complex hyperbolic spaces*, IV Encuentro Conjunto RSME-SMM, Valladolid, Spain, 19-22 June 2017. - V. Sanmartín-López:
*Description and classification of isoparametric submanifolds in the complex hyperbolic space*, Differential Geometry, Bedlewo, Poland, 18-24 June 2017. - J. C. Díaz-Ramos:
*Cohomogeneity one actions on Minkowski spaces*, 8th International Meeting on Lorentzian Geometry, Málaga, Spain, 20-23 September 2016. - M. Domínguez-Vázquez:
*Isoparametric submanifolds of complex space forms*, XXV International Fall Workshop on Geometry and Physics, Madrid, Spain, 29 August - 2 September 2016. - V. Sanmartín-López:
*Anti-De Sitter and complex hyperbolic spaces: their isoparametric hypersurfaces*, Differential Geometry and its Applications, Brno, Czech Republic, 11-15 July 2016. - J. C. Díaz-Ramos:
*Isoparametric hypersurfaces in complex hyperbolic spaces*, Second joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies, Logroño, Spain, 6-8 June 2016. - J. C. Díaz-Ramos:
*Non-proper cohomogeneity one actions on Minkowski spaces*, First joint meeting Brazil-Spain in Mathematics, Fortaleza, Brazil, 7-10 December 2015. - M. Domínguez-Vázquez:
*Classification of isoparametric hypersurfaces in complex hyperbolic spaces*, First joint meeting Brazil-Spain in Mathematics, Fortaleza, Brazil, 7-10 December 2015. - C. Vidal-Castiñeira:
*Hipersuperficies de cohomogeneidad uno en los planos proyectivo e hiperbólico complejos*, Congreso de Jóvenes Investigadores, Murcia, Spain, 7-11 September 2015. - V. Sanmartín-López:
*Clasificación de hipersuperficies isoparamétricas en el espacio hiperbólico complejo*, Congreso de Jóvenes Investigadores, Murcia, Spain, 7-11 September 2015. - C. Vidal-Castiñeira:
*Cohomegeneity one hypersurfaces in the complex projective and hyperbolic planes*, XXIV International Fall Workshop on Geometry and Physics, Zaragoza, Spain, 1-4 September 2015. - V. Sanmartín-López:
*Hipersuperficies isoparamétricas en el espacio hiperbólico complejo*, XI Encuentro Andaluz de Geometría, Sevilla, Spain, . - J. C. Díaz-Ramos:
*Polar actions on symmetric spaces*, RSME Conference 2015, Granada, Spain, 2-6 February 2015. - M. Domínguez-Vázquez:
*Two construction methods of isoparametric hypersurfaces in noncompact symmetric spaces*, RSME Conference 2015, Granada, Spain, 2-6 February 2015. - C. Vidal-Castiñeira:
*Hipersuperficies con dos curvaturas principales en CP^2 y CH^2*, RSME Conference 2015, Granada, Spain, 2-6 February 2015. - M. Domínguez-Vázquez:
*Classification of isoparametric foliations on complex projective spaces*, Geometry and Dynamics of Foliations, Madrid, Spain, 1 September - 5 August 2014. - M. Domínguez-Vázquez:
*Polar actions on complex hyperbolic spaces II*, XVIII Escola de Geometria Diferencial, Brasilia, Brazil, 28 July - 1 August 2014. - J. C. Díaz-Ramos:
*Real hypersurfaces in complex space forms*, The second Japanese-Spanish workshop on Differential Geometry, Tokyo, Japan, 5-10 February 2014. - M. Domínguez-Vázquez:
*Isoparametric submanifolds of complex projective spaces*, The second Japanese-Spanish workshop on Differential Geometry, Tokyo, Japan, 5-10 February 2014. - M. Domínguez-Vázquez:
*Some interesting families of non-Hopf real hypersurfaces in complex space forms*, 17th International Workshop on Differential Geometry, Daejeon, South Korea, 30 September - 2 October 2013. - M. Domínguez-Vázquez:
*Familias isoparamétricas en espacios de Damek-Ricci*, Congreso de Jóvenes Investigadores de la RSME, Seville, Spain, 16-20 September 2013. - M. Domínguez-Vázquez:
*The horospherical decomposition and new isoparametric hypersurfaces in noncompact symmetric spaces*, Geometric Analysis Seminar GAS-UV13, Valencia, Spain, 4-7 September 2013. - J. C. Díaz-Ramos:
*Polar actions on complex space forms II*, The 16th International Workshop on Differential Geometry & Related Topics, Daegu, South Korea, 1-3 November 2012. - J. C. Díaz-Ramos:
*Polar actions on complex space forms I*, Pre-joint Meeting of The 16th International Workshop on Differential Geometry, Daegu, South Korea, . - M. Domínguez-Vázquez:
*Polar foliations for beginners*, Geometry Meeting 2012, Ferrol, Spain, . - M. Domínguez-Vázquez:
*On the existence of inhomogeneous isoparametric foliations of higher codimension on complex projective spaces*, Conference on Pure and Applied Differential Geometry PADGE 2012, Leuven, Belgium, 27-30 August 2012. - J. C. Díaz-Ramos:
*Polar actions on complex hyperbolic spaces*, Differential Geometry Days, Santiago de Compostela, Spain, 27-29 June 2012. - M. Domínguez-Vázquez:
*Isoparametric foliations on complex projective spaces*, Differential Geometry Days, Santiago de Compostela, Spain, 27-29 June 2012. - J. C. Díaz-Ramos:
*Polar actions on noncompact symmetric spaces*, 2nd European Young and Mobile Workshop, Geometric Analysis and Partial Differential Equations, Granada, Spain, 28-30 November 2011. - J. C. Díaz-Ramos:
*Polar actions on symmetric spaces of noncompact type*, VI International Meeting on Lorentzian Geometry, Granada, Spain, 6-9 September 2011. - M. Domínguez-Vázquez:
*Families of parallel CMC hypersurfaces in noncompact rank-one symmetric spaces*, VI International Meeting on Lorentzian Geometry, Granada, Spain, 6-9 September 2011. - M. Domínguez-Vázquez:
*Principal curvatures of isoparametric hypersurfaces in complex hyperbolic spaces*, Conference in Geometry and Global Analysis, Santiago de Compostela, Spain, 13-17 December 2010. - M. Domínguez-Vázquez:
*On isoparametric hypersurfaces in complex hyperbolic spaces*, International Meeting on Differential Geometry, Córdoba, Spain, 15-17 November 2010. - M. Domínguez-Vázquez:
*O problema da clasificación das hipersuperficies isoparamétricas*, Encontro de Xeometría 2010, Ferrol, Spain, . - M. Domínguez-Vázquez:
*Real hypersurfaces with constant principal curvatures in complex space forms*, DGA 2010 - Differential Geometry and its Applications, Brno, Czech Republic, 27-31 August 2010. - J. C. Díaz-Ramos:
*Homogeneous hyperpolar foliations on symmetric spaces of noncompact type*, Workshop on Hypersurface Geometry and Integrable Systems, Sendai, Japan, 24-27 August 2010. - M. Domínguez-Vázquez:
*Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces*, Workshop on Hypersurface Geometry and Integrable Systems, Sendai, Japan, 24-27 August 2010. - J. C. Díaz-Ramos:
*Acciones hiperpolares en espacios simétricos no compactos*, Geometry Meeting 2009, Ferrol, Spain, 29-30 October 2009. - J. C. Díaz-Ramos:
*Hyperpolar actions on noncompact symmetric spaces*, II Congreso latinoamericano de grupos de Lie en geometría, La Falda, Argentina, 25-29 September 2008. - J. C. Díaz-Ramos:
*Hypersurfaces with constant principal curvatures in the complex hyperbolic space*, VIII International Colloquium on Differential Geometry, Santiago de Compostela, Spain, 7-11 July 2008. - J. C. Díaz-Ramos:
*A characterisation of real hypersurfaces with constant principal curvatures*, Differential Geometry, Bedlewo, Poland, 22-28 June 2008. - J. C. Díaz-Ramos:
*Homogeneous hypersurfaces in the complex hyperbolic space*, Irish Geometry Conference, Galway, Ireland, 24-25 May 2007.

### Invited seminar talks

- V. Sanmartín-López:
*Description of isoparametric hypersurfaces in the complex hyperbolic space*, Seminario de Geometría, Madrid, Spain, . - V. Sanmartín-López:
*On the classification of isoparametric hypersurfaces*, Geometry Seminar, Cologne, Germany, . - J. C. Díaz-Ramos:
*Isoparametric hypersurfaces in complex space forms*, Department of Mathematics Seminar, Stuttgart, Germany, . - V. Sanmartín-López:
*Isoparametric hypersurfaces in the anti-De Sitter space and the complex hyperbolic space*, Geometry Seminar, London, United Kingdom, . - M. Domínguez-Vázquez:
*Isoparametric hypersurfaces in overdetermined PDE problems*, Seminario Vidal Abascal, Santiago de Compostela, España, . - M. Domínguez-Vázquez:
*Clasificación das foliacións polares no espazo proxectivo cuaterniónico*, Seminario Vidal Abascal, Santiago de Compostela, España, . - J. C. Díaz-Ramos:
*On the classification of polar actions on complex hyperbolic spaces*, Workshop on submanifolds in complex space forms, London, United Kingdom, . - C. Vidal-Castiñeira:
*Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes*, Workshop on submanifolds in complex space forms, London, United Kingdom, . - V. Sanmartín-López:
*On the classification of isoparametric hypersurfaces in complex hyperbolic spaces*, Workshop on submanifolds in complex space forms, London, United Kingdom, . - M. Domínguez-Vázquez:
*Polar foliations on quaternionic projective spaces*, Geometry Seminar, São Paulo, Brazil, . - V. Sanmartín-López:
*Isoparametric families of hypersurfaces in complex hyperbolic spaces*, Differential Geometry Seminar, Rio de Janeiro, Brazil, . - J. C. Díaz-Ramos:
*Cohomogeneity one actions on the Minkowski space*, Seminar, Rio de Janeiro, Brazil, . - J. C. Díaz-Ramos:
*Foliaciones polares e isoparamétricas*, Department of Mathematics Seminar, Bilbao, Spain, . - M. Domínguez-Vázquez:
*Polar foliations on complex projective space*, Seminar, Rio de Janeiro, Brazil, . - M. Domínguez-Vázquez:
*Introduction to isoparametric foliations*, Seminar, Sao Paulo, Brazil, 23-30 June 2014. - M. Domínguez-Vázquez:
*Inhomogeneous isoparametric foliations and prime numbers*, Seminar, Sao Paulo, Brazil, . - M. Domínguez-Vázquez:
*Homogeneity of isoparametric submanifolds and prime numbers*, Seminar, Brasilia, Brazil, . - M. Domínguez-Vázquez:
*Hypersurfaces with two principal curvatures in nonflat complex space forms*, Department of Mathematics Seminar, IMPA, Rio de Janeiro, Brazil, . - M. Domínguez-Vázquez:
*A geometric characterization via prime numbers*, Department of Mathematics Seminar, Universidade Federal Fluminense, Rio de Janeiro, Brazil, . - M. Domínguez-Vázquez:
*Inhomogeneous isoparametric foliations*, Department of Mathematics Seminar, London, United Kingdom, . - M. Domínguez-Vázquez:
*The classification problem of isoparametric foliations*, ICMAT Seminar, Madrid, Spain, . - M. Domínguez-Vázquez:
*Inhomogeneous isoparametric families on complex projective spaces*, Department of Mathematics Seminar, Stuttgart, Germany, . - J. C. Díaz-Ramos:
*A classification of polar actions on complex hyperbolic spaces*, Department of Mathematics Seminar, Cologne, Germany, . - J. C. Díaz-Ramos:
*Isoparametric hypersurfaces in complex space forms*, Department of Mathematics Seminar, Darmstadt, Germany, . - J. C. Díaz-Ramos:
*Foliaciones polares homogéneas en espacios hiperbólicos complejos*, Xornada de Xeometría (Seminario Vidal Abascal), Santiago de Compostela, Spain, . - M. Domínguez-Vázquez:
*Homogeneous and isoparametric hypersurfaces in non-compact rank-one symmetric spaces*, Department of Mathematics Seminar, Cologne, Germany, . - J. C. Díaz-Ramos:
*Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms*, Department of Mathematics Seminar, Cologne, Germany, . - J. C. Díaz-Ramos:
*Homogeneous hyperpolar foliations on symmetric spaces of noncompact type*, Department of Mathematics Seminar, Cologne, Germany, . - J. C. Díaz-Ramos:
*Hyperpolar homogeneous foliations on noncompact symmetric spaces*, Department of Mathematics Seminar, Augsburg, Germany, . - J. C. Díaz-Ramos:
*Hypersurfaces with constant principal curvatures in the complex hyperbolic space*, Hiroshima University Topology-Geometry Seminar, Hiroshima, Japan, . - J. C. Díaz-Ramos:
*Hipersuperficies homoxéneas no espazo hiperbólico real*, Seminario de Xeometría e Topoloxía, Santiago de Compostela, Spain, .

### Posters

- J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira:
*Real hypersurfaces with two constant principal curvatures in complex projective and hyperbolic planes*, XXIII International Fall Workshop on Geometry and Physics, Granada, Spain, 1-6 September 2014. - M. Domínguez-Vázquez:
*Some results on isoparametric hypersurfaces in nonflat complex space forms*, XX International Fall Workshop on Geometry and Physics, Madrid, Spain, 31 August - 3 September 2011. - J. C. Díaz-Ramos, M. Domínguez-Vázquez:
*Principal curvatures of isoparametric hypersurfaces in complex hyperbolic spaces II*, Conference in Geometry and Global Analysis, Santiago de Compostela, Spain, 13-17 December 2010. - J. C. Díaz-Ramos, M. Domínguez-Vázquez:
*Hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces*, Lectures on Lorentzian Geometry, Granada, Spain, 8-11 February 2010.