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 geométricas (Symmetry, curvature and geometric equations), Agencia Estatal de Investigación PID2019-105138GB-C21 (Spain) (62920 euro), 1 June 2020 - 31 May 2023. Principal investigator: José Carlos Díaz Ramos, Eduardo García Río.
• Simetría, curvatura y ecuaciones diferenciales en geometría (Symmetry, curvature and differential equations in geometry), Agencia Estatal de Investigación MTM2016-75897-P (Spain) (72963 euro), 30 December 2016 - 29 December 2019. Principal investigator: José Carlos Díaz Ramos, Eduardo García Río.
• Homoxeneidade e curvatura de variedades e subvariedades (Homogeneity and curvature of manifolds and submanifolds), Galician government ED431F 2017/03 (Spain) (40000 euro), 1 January 2017 - 30 November 2018. Principal investigator: José Carlos Díaz Ramos.
• 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:

• Alberto Rodríguez Vázquez: Homogeneous hypersurfaces and totally geodesic submanifolds, Ph.D. thesis, University of Santiago de Compostela, 30 September 2022.
• Olga Pérez Barral: Ruled hypersurfaces and homogeneous submanifolds in semi-Riemannian manifolds, Ph.D. thesis, University of Santiago de Compostela, 11 December 2020.
• Víctor Sanmartín López: Homogeneous submanifolds and isoparametric hypersurfaces in symmetric spaces of noncompact type, Ph.D. thesis, University of Santiago de Compostela, 19 July 2019.
• Cristina Vidal Castiñeira: Submanifolds in complex projective and hyperbolic planes, Ph.D. thesis, University of Santiago de Compostela, 22 July 2016.
• Mohammad Javad Vanaei: Cohomogeneity one Lorentzian Manifolds of Constant Curvature, Ph.D. thesis, Tarbiat Modares University, 25 August 2015.
• Miguel Domínguez Vázquez: Isoparametric foliations and polar actions on complex space forms, Ph.D. thesis, University of Santiago de Compostela, 5 March 2013.

I have also been advisor of the following research works:

• Juan Manuel Lorenzo Naveiro: Homogeneous polar foliations in symmetric spaces (Foliaciones polares homogéneas en espacios simétricos), master thesis, University of Santiago de Compostela, 15 July 2021.
• Alberto Rodríguez Vázquez: Hipersuperficies con curvaturas principais constantes en variedades Kähler con curvatura seccional holomorfa constante (Hypersurfaces with constant principal curvatures in Kähler manifolds of constant holomorphic sectional curvature), master thesis, University of Santiago de Compostela, 17 July 2018.
• Olga Pérez Barral: Acciones isométricas en el espacio-tiempo de Minkowski (Isometric actions on Minkowski spacetimes), master thesis, University of Santiago de Compostela, 20 July 2017.
• Víctor Sanmartín López: Hipersuperficies isoparamétricas en el espacio hiperbólico complejo (Isoparametric hypersurfaces in the complex hyperbolic space), master thesis, University of Santiago de Compostela, 23 July 2014.
• 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, 22 July 2013.
• 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, 8 July 2010.

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, J. M. Lorenzo-Naveiro: Codimension two polar homogeneous foliations on symmetric spaces of noncompact type, arXiv:2302.08339 [math.DG].
• M. Domínguez-Vázquez, V. Sanmartín-López: Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank, arXiv:2109.03850 [math.DG].
• J. Berndt, V. Sanmartín-López: Submanifolds with constant principal curvatures in Riemannian symmetric spaces, to appear in Comm. Anal. Geom.
• A. Rodríguez-Vázquez, C. Olmos: Hopf fibrations and totally geodesic submanifolds, arXiv:2302.11711 [math.DG].
• A. Rodríguez-Vázquez: Totally geodesic submanifolds in products of rank one symmetric spaces, arXiv:2205.14720 [math.DG].
• A. Rodríguez-Vázquez, A. Kollross: Totally geodesic submanifolds in exceptional symmetric spaces, to appear in Adv. Math.
• J. M. Lorenzo-Naveiro, I. Solonenko: Sections of polar actions, arXiv:2111.05280 [math.DG].
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, T. Otero-Casal: Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank, Adv. Math. 428 (2023), 109165.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, O. Pérez-Barral: Homogeneous CR submanifolds of complex hyperbolic spaces, Publ. Mat. 67 (2023), 891-912.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, T. Hashinaga: Homogeneous Lagrangian foliations on complex space forms, Proc. Amer. Math. Soc. 151 (2023), 823-833.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, A. Rodríguez-Vázquez: Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces, J. Reine Angew. Math. 779 (2021), 189-222.
• M. Domínguez-Vázquez, O. Pérez-Barral: Existence and uniqueness of inhomogeneous ruled hypersurfaces with shape operator of constant norm in the complex hyperbolic space, Internat. J. Math. 32 (2021), 2150049, 4 pp.
• M. Domínguez-Vázquez, V. Sanmartín-López, H. Tamaru: Codimension one Ricci soliton subgroups of solvable Iwasawa groups, J. Math. Pures Appl. 152 (2021), 69-93.
• M. Domínguez-Vázquez, J. M. Manzano: Isoparametric surfaces in E(κ, τ)-spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), 269-285.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, A. Kollross: On homogeneous manifolds whose isotropy actions are polar, Manuscripta Math. 161 (2020), 15-34.
• O. Pérez-Barral: Some problems on ruled real hypersurfaces in non flat complex space forms, Results Math., 75, 167 (2020).
• M. Domínguez-Vázquez, O. Pérez-Barral: Ruled hypersurfaces with constant mean curvature in complex space forms, J. Geom. Phys. 144 (2019), 121-125.
• A. Rodríguez-Vázquez: A nonisoparametric hypersurface with constant principal curvatures, Proc. Amer. Math. Soc. 147 (2019), 5417-5420.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes, Ann. Mat. Pura Appl. 197 (2018), 469-486.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Isoparametric submanifolds in two-dimensional complex space forms, Ann. Global Anal. Geom. 53 (2018), 205-216.
• M. Domínguez-Vázquez, C. Gorodski: Polar foliations on quaternionic projective spaces, Tohoku Math. J. 70 (2018), 353-375.
• 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, A. Kollross: Polar actions on complex hyperbolic spaces, Math. Z. 287 (2017), 1183-1213.
• 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.
• V. Sanmartín-López: Spacelike isoparametric hypersurfaces, Differential Geom. Appl., 54, Part A (2017), 53-58.
• 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, T. Otero-Casal: Homogeneous hypersurfaces in symmetric spaces, New Trends in Geometric Analysis, Spanish Network of Geometric Analysis 2007-2021, RSME Springer Series, Springer, 2023, 141-190.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, V. Sanmartín-López: Submanifold geometry in symmetric spaces of noncompact type, Sao Paulo J. Math. Sci., (Special Section: An Homage to Manfredo P. do Carmo), 2021, 75-110.
• J. C. Díaz-Ramos, M. Domínguez-Vázquez, V. Sanmartín-López: Anti-De Sitter spacetimes and isoparametric hypersurfaces in complex hyperbolic spaces, Lorentzian Geometry and Related Topics, Springer Proceedings in Mathematics & Statistics 211, GeLoMa 2016, Málaga, Spain, September 20-23, Springer, 2017, 87-101.
• 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 Lagrangian foliations in complex spaces forms, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• M. Domínguez-Vázquez: Extension of submanifolds in symmetric spaces of noncompact type, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• V. Sanmartín-López: Codimension one Ricci solitons in Iwasawa groups, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• A. Rodríguez-Vázquez: Totally geodesic submanifolds in products of rank one symmetric spaces, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• T. Otero-Casal: Homogeneous hypersurfaces of symmetric spaces of noncompact type, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• J. M. Lorenzo-Naveiro: Polar actions on some noncompact symmetric spaces, Workshop on Manifolds with Symmetries, Stuttgart, Germany, 28-30 March 2022.
• J. C. Díaz-Ramos: Cohomogeneity one actions on rank one symmetric spaces of noncompact type, Geometry of symmetric spaces and group actions, online, Japan, 17-19 February 2022.
• M. Domínguez-Vázquez: Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank, Geometry of symmetric spaces and group actions, online, Japan, 17-19 February 2022.
• A. Rodríguez-Vázquez: Totally geodesic submanifolds and Kähler angle in symmetric spaces, Symmetry and Shape, Santiago de Compostela, Spain, 15-18 October 2021.
• T. Otero-Casal: Cohomogeneity one actions on irreducible and reducible symmetric spaces of noncompact type, Symmetry and Shape, Santiago de Compostela, Spain, 15-18 October 2021.
• J. M. Lorenzo-Naveiro: Polar homogeneous foliations on SL(3,R)/SO(3), Symmetry and Shape, Santiago de Compostela, Spain, 15-18 October 2021.
• A. Rodríguez-Vázquez: Symmetric spaces: holonomy and totally geodesic submanifolds, PG Forum, Lancaster, United Kingdom, 8 March 2021.
• O. Pérez-Barral: Subvariedades CR homogéneas en el espacio hiperbólico complejo, Congreso de Jóvenes Investigadores RSME, Castelló, Spain, 27-31 January 2020.
• A. Rodríguez-Vázquez: Isoparametricidad e isospectralidad del operador forma, Congreso de Jóvenes Investigadores RSME, Castelló, Spain, 27-31 January 2020.
• A. Rodríguez-Vázquez: Homogeneous and inhomogeneous isoparametric hypersurfaces in quaternionic hyperbolic spaces, Workshop on Geometric Analysis IEMATH-GR, Granada, Spain, 5 December 2019.
• J. C. Díaz-Ramos: Homogeneous submanifolds in complex space forms, Symmetry and shape, Celebrating the 60th birthday of Prof. J. Berndt, Santiago de Compostela, Spain, 28-31 October 2019.
• O. Pérez-Barral: Homogeneous CR submanifolds in complex hyperbolic spaces, Symmetry and shape: Celebrating the 60th birthday of Prof. J. Berndt, Santiago de Compostela, Spain, 28-31 October 2019.
• M. Domínguez-Vázquez: Cohomogeneity one isometric actions on quaternionic hyperbolic spaces, Encuentro de Topología 2019, Santiago de Compostela, Spain, 18-19 October 2019.
• A. Rodríguez-Vázquez: A non isoparametric hypersurface with constant principal curvatures, Differential Geometry and its Applications, Hradrec Králové, Czech Republic, 1-7 September 2019.
• M. Domínguez-Vázquez: Homogeneous and inhomogeneous isoparametric hypersurfaces in symmetric spaces of noncompact type, Workshop on the Isoparametric Theory, Beijing , China, 2-6 June 2019.
• M. Domínguez-Vázquez: Isoparametric surfaces in E(κ, τ)-spaces, Encuentro de la Red Española de Análisis Geométrico, Murcia, Spain, 21 April 2019.
• J. C. Díaz-Ramos: Cohomogeneity one actions on Minkowski spaces, II joint meeting Brazil-Spain in Mathematics, Cádiz, Spain, 11-14 December 2018.
• M. Domínguez-Vázquez: Nonlinear overdetermined boundary problems in Riemannian manifolds, II joint meeting Brazil-Spain in Mathematics, Cádiz, Spain, 11-14 December 2018.
• M. Domínguez-Vázquez: Subvariedades e accións isométricas en espazos simétricos de tipo non compacto, Seminario Vidal Abascal, Santiago de Compostela, Spain, 29-31 October 2018.
• M. Domínguez-Vázquez: Isoparametric surfaces in homogeneous 3-spaces, Preliminary talks of the ICM Satellite Conference Modern Trends in Differential Geometry, São Paulo, Brazil, 19 July 2018.
• M. Domínguez-Vázquez: Cohomogeneity one actions on symmetric spaces of noncompact type, Seminar, Rio de Janeiro, Brazil, 12 July 2018.
• O. Pérez-Barral: Ruled hypersurfaces with constant mean curvature in complex space forms are minimal, First BYMAT Conference: Bringing Young Mathematicians Together, Madrid, Spain, 7-9 May 2018.
• V. Sanmartín-López: Isospectral submanifolds in non-compact symmetric spaces, Hiroshima Differential Geometry Day 2018, Hiroshima, Japan, 21 April 2018.
• V. Sanmartín-López: Homogeneous and isoparametric hypersurfaces in complex hyperbolic spaces, Geometry of Submanifolds and Integrable Systems, Osaka, Japan, 26-30 March 2018.
• M. Domínguez-Vázquez: Isoparametric hypersurfaces in the complex hyperbolic space, Primer Encuentro Conjunto RSME-UMA, Buenos Aires, Argentina, 11-15 December 2017.
• 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.
• M. Domínguez-Vázquez: Submanifolds and isometricactions on symmetric spaces of noncompact type, Seminar, São Paulo, Brazil, 27 March - 9 April 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, 15 May 2015.
• 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, 31 October 2012.
• M. Domínguez-Vázquez: Polar foliations for beginners, Geometry Meeting 2012, Ferrol, Spain, 26 October 2012.
• 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, 7 October 2010.
• 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

• J. C. Díaz-Ramos: Cohomogeneity one actions on quaternionic hyperbolic spaces, Symmetric spaces seminar, online, United Kingdom, 4 May 2022.
• M. Domínguez-Vázquez: Cohomogeneity one actions and inhomogeneous isoparametric hypersurfaces in quaternionic hyperbolic spaces, Differential Geometry Seminar, online, Germany, 15 June 2020.
• A. Rodríguez-Vázquez: Homogeneous and inhomogeneous isoparametric hypersurfaces in symmetric spaces, Junior Geometry Seminar KCL/UCL, London, United Kingdom, 12 March 2020.
• M. Domínguez-Vázquez: On isoparametric hypersurfaces in noncompact symmetric spaces, Seminar, Norman, United States, 3 October 2018.
• M. Domínguez-Vázquez: Overdetermined elliptic problems in Riemannian manifolds, Seminar, South Bend, United States, 20 September 2018.
• M. Domínguez-Vázquez: Isoparametric surfaces in homogeneous 3-manifolds, Seminar, St. Louis, United States, 14 September 2018.
• M. Domínguez-Vázquez: Polar foliations on symmetric spaces, Colloquium, St. Louis, United States, 13 September 2018.
• M. Domínguez-Vázquez: Isoparametric surfaces in homogeneous spaces, Seminario Vidal Abascal, Santiago de Compostela, Spain, 7 June 2018.
• V. Sanmartín-López: Description of isoparametric hypersurfaces in the complex hyperbolic space, Seminario de Geometría, Madrid, Spain, 7 June 2017.
• V. Sanmartín-López: On the classification of isoparametric hypersurfaces, Geometry Seminar, Cologne, Germany, 1 June 2017.
• J. C. Díaz-Ramos: Isoparametric hypersurfaces in complex space forms, Department of Mathematics Seminar, Stuttgart, Germany, 26 January 2017.
• V. Sanmartín-López: Isoparametric hypersurfaces in the anti-De Sitter space and the complex hyperbolic space, Geometry Seminar, London, United Kingdom, 30 November 2016.
• M. Domínguez-Vázquez: Isoparametric hypersurfaces in overdetermined PDE problems, Seminario Vidal Abascal, Santiago de Compostela, Spain, 21 July 2016.
• M. Domínguez-Vázquez: Clasificación das foliacións polares no espazo proxectivo cuaterniónico, Seminario Vidal Abascal, Santiago de Compostela, Spain, 3 March 2016.
• J. C. Díaz-Ramos: On the classification of polar actions on complex hyperbolic spaces, Workshop on submanifolds in complex space forms, London, United Kingdom, 21 January 2016.
• C. Vidal-Castiñeira: Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes, Workshop on submanifolds in complex space forms, London, United Kingdom, 21 January 2016.
• V. Sanmartín-López: On the classification of isoparametric hypersurfaces in complex hyperbolic spaces, Workshop on submanifolds in complex space forms, London, United Kingdom, 21 January 2016.
• M. Domínguez-Vázquez: Polar foliations on quaternionic projective spaces, Geometry Seminar, São Paulo, Brazil, 23 October 2015.
• V. Sanmartín-López: Isoparametric families of hypersurfaces in complex hyperbolic spaces, Differential Geometry Seminar, Rio de Janeiro, Brazil, 8 October 2015.
• J. C. Díaz-Ramos: Cohomogeneity one actions on the Minkowski space, Seminar, Rio de Janeiro, Brazil, 2 June 2015.
• J. C. Díaz-Ramos: Foliaciones polares e isoparamétricas, Department of Mathematics Seminar, Bilbao, Spain, 22 May 2015.
• M. Domínguez-Vázquez: Polar foliations on complex projective space, Seminar, Rio de Janeiro, Brazil, 9 October 2014.
• 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, 30 May 2014.
• M. Domínguez-Vázquez: Homogeneity of isoparametric submanifolds and prime numbers, Seminar, Brasilia, Brazil, 31 March 2014.
• M. Domínguez-Vázquez: Hypersurfaces with two principal curvatures in nonflat complex space forms, Department of Mathematics Seminar, IMPA, Rio de Janeiro, Brazil, 16 January 2014.
• M. Domínguez-Vázquez: A geometric characterization via prime numbers, Department of Mathematics Seminar, Universidade Federal Fluminense, Rio de Janeiro, Brazil, 7 November 2013.
• M. Domínguez-Vázquez: Inhomogeneous isoparametric foliations, Department of Mathematics Seminar, London, United Kingdom, 7 May 2013.
• M. Domínguez-Vázquez: The classification problem of isoparametric foliations, ICMAT Seminar, Madrid, Spain, 11 February 2013.
• M. Domínguez-Vázquez: Inhomogeneous isoparametric families on complex projective spaces, Department of Mathematics Seminar, Stuttgart, Germany, 19 June 2012.
• J. C. Díaz-Ramos: A classification of polar actions on complex hyperbolic spaces, Department of Mathematics Seminar, Cologne, Germany, 10 June 2012.
• J. C. Díaz-Ramos: Isoparametric hypersurfaces in complex space forms, Department of Mathematics Seminar, Darmstadt, Germany, 12 December 2011.
• J. C. Díaz-Ramos: Foliaciones polares homogéneas en espacios hiperbólicos complejos, Xornada de Xeometría (Seminario Vidal Abascal), Santiago de Compostela, Spain, 21 July 2011.
• M. Domínguez-Vázquez: Homogeneous and isoparametric hypersurfaces in non-compact rank-one symmetric spaces, Department of Mathematics Seminar, Cologne, Germany, 19 April 2011.
• J. C. Díaz-Ramos: Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms, Department of Mathematics Seminar, Cologne, Germany, 16 November 2010.
• J. C. Díaz-Ramos: Homogeneous hyperpolar foliations on symmetric spaces of noncompact type, Department of Mathematics Seminar, Cologne, Germany, 2 July 2010.
• J. C. Díaz-Ramos: Hyperpolar homogeneous foliations on noncompact symmetric spaces, Department of Mathematics Seminar, Augsburg, Germany, 8 January 2010.
• J. C. Díaz-Ramos: Hypersurfaces with constant principal curvatures in the complex hyperbolic space, Hiroshima University Topology-Geometry Seminar, Hiroshima, Japan, 8 April 2008.
• J. C. Díaz-Ramos: Hipersuperficies homoxéneas no espazo hiperbólico real, Seminario de Xeometría e Topoloxía, Santiago de Compostela, Spain, 23 March 2006.

Posters

• A. Rodríguez-Vázquez: Non-Hopf hypersurfaces with constant principal curvatures in complex projective space and complex hyperbolic space, Congreso Bienal de la RSME, Santander, Spain, 4-8 February 2019.
• O. Pérez-Barral: Ruled hypersurfaces with constant mean curvature in complex space forms, XXV Encuentro de Topología, Barcelona, Spain, 19-20 October 2018.
• A. Rodríguez-Vázquez: Hypersurfaces with constant principal curvatures in complex hyperbolic spaces, XXV Encuentro de Topología, Barcelona, Spain, 19-20 October 2018.
• 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.