Biharmonic Hypersurfaces in Projective Spaces Revisited

Year 2025, Volume: 18 Issue: 2, 293 - 334

Jun-ichi Inoguchi , Toru Sasahara

https://doi.org/10.36890/iejg.1665105

Abstract

We study biharmonic homogeneous real hypersurfaces in complex projective space and quaternion projective space. We provide a classification of biharmonic homogeneous real hypersurfaces in quaternion projective space. We also classify pseudo-harmonic, subelliptic biharmonic, and Levi-harmonic homogeneous Hopf hypersurfaces in complex space forms.

Keywords

Biharmonic maps , complex projective space , quaternion projective space , Riemannian symmetric spaces , hypersurfaces , CR-manifolds; pseudo-harmonic map , subelliptic biharmonic map , Levi-harmonic map

References

• Adachi, T., Bao, T., Maeda, S.: Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form. Hokkaido Math. J. 43 (1), 137-150 (2014).
• Adachi, T., Kameda, M., Maeda, S.: Real hypersurfaces which are contact in a nonflat complex space form. Hokkaido Math. J. 40, 205-217 (2011).
• Adachi, T., Maeda, S., Udagawa, S.: Ruled real hypersurfaces in a nonflat quaternionic space form. Mh. Math. 145 (3), 179-190 (2005).
• Barletta, E., Dragomir S., Urakawa, H.: Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds. Indiana Univ. Math. J. 50 (2), 719-746 (2001).
• Bejancu, A.: CR submanifolds of a Kaehler manifold. I. Proc. Amer. Math. Soc. 69 (1), 135-142 (1978).
• Berndt, J.: Real hypersurfaces with constant principal curvatures in complex space forms. Geometry and Topology of Submanifolds, III, (M. Boyom, J. M. Morvan and L. Verstraelen eds), World Scientific, pp. 10-19, (1999).
• Berndt, J.: Real hypersurfaces in quaternionic space forms. J. Reine Angew. Math. 419, 9-26 (1991).
• Berndt, J., Díaz-Ramos, J. C.: Homogeneous hypersurfaces in complex hyperbolic spaces. Geom. Dedicata 138, 129-150 (2009).
• Berndt, J., Suh, Y. J.: Contact hypersurfaces in Kähler manifolds. Proc. Amer. Math. Soc. 143, 2637-2649 (2015).
• Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit. Tôhoku Math. J. (2) 56, 163-177 (2004).
• Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds. 2nd. ed., Progress in Math. 203, Birkhäuser, Boston, Basel, Berlin, (2010).
• Blair, D. E., Chen, B.-Y.: On CR-submanifolds of Hermitian manifolds. Israel J. Math. 34 (4), 353-363 (1979).
• Blair, D. E., Dragomir, S.: Pseudo-Hermitian geometry on contact Riemannian manifolds. Rend. Mat. Appl., VII. Ser. 22, 275-341 (2002).
• Blair, D. E., Koufogiorgos, Th., Papantoniou, B. J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189-214 (1995).
• Burns Jr, D., Shnider, S.: Spherical hypersurfaces in complex manifolds. Invent. Math. 33 (3), 223-246 (1976).
• Brandão, C.: Biharmonic submanifolds of the quaternionic projective space. J. Geom. Phys. 206, Article number 105310, (2024).
• Cecil, T. E.: Geometric applications of critical point theory to submanifolds of complex projective spaces. Nagoya Math. J. 55, 5-31 (1974).
• Cecil, T. E., Ryan, P. J.: Focal sets and real hypersurfaces in complex projective spaces. Trans. Amer. Math. Soc. 269, 481-499 (1982).
• Cecil, T. E., Ryan, P. J.: Geometry of Hypersurfaces. Springer Monogr. Math. Springer, New York, (2015).
• Chen, B.-Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific, (1984).
• Chern, S. S., Moser, J. K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). Erratum: 150 (3-4), 297 (1983).
• Cho, J. T., Inoguchi, J.: Contact metric hypersurfaces in complex space forms. Proceedings of the workshop on Differential Geometry of Submanifolds and its Related Topics Saga (August 4-6, 2012), World Scientific, 2014, 87-97 (2014).
• Cho, J. T., Inoguchi, J. Lee, J.-E.: Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry. Abh. Math. Semin. Univ. Hambg. 79, 113-133 (2009).
• Cho, J. T., Kimura, M.: Spherical CR-symmetric hypersurfaces in Hermitian symmetric spaces. Illinois J. Math. 67 (3), 547-562 (2023).
• Chong, T., Dong, Y., Ren, Y., Yang, G.: On harmonic and pseudoharmonic maps. Nagoya Math. J. 234, 170-210 (2019).
• Corlette, K.: Archimedean superrigidity and hyperbolic geometry. Ann. Math. (2) 135 (1), 165-182 (1992).
• D’Atri, J. E.: Certain isoparametric families of hypersurfaces in symmetric spaces. J. Differential Geom. 14, 21-40 (1979).
• Dileo, G., Lotta, A.: A classification of spherical symmetric CR manifolds. Bull. Aust. Math. Soc. 80 (2), 251-274 (2009).
• Dong, Y.: Eells-Sampson type theorems for subelliptic harmonic maps from sub-Riemannian manifolds. J. Geom. Anal. 31 (4), 3608-3655 (2021).
• Dragomir, S., Montaldo, S.: Subelliptic biharmonic maps. J. Geom. Anal.24 (1), 223-245 (2014).
• Dragomir, S, Perrone, D.: Levi harmonic maps of contact Riemannian manifolds. J. Geom. Anal.24 (3), 1233-1275 (2014).
• Eells, J., Sampson, J. H.: Variational theory in fibre bundles. Proc. United States-Japan Semin. Differ. Geom., Kyoto 1965, 22-23, (1966).
• Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps. Regional Conference Series in Mathematics 50, Amer. Math. Soc. (1983).
• Eells, J., Wood, J. C.: Restrictions on harmonic maps of surfaces. Topology 15, 263-266 (1976).
• Ejiri, N.: A generalization of minimal cones. Trans. Amer. Math. Soc. 276, 347-360 (1983).
• Enoyoshi, K.: Principal curvatures of homogeneous hypersurfaces in a Grassmann manifold G+ 3 (ImO) by the G2-action. Tokyo J. Math. 42 (2), 571-584 (2019).
• Erdem, S.: On harmonicity of holomorphic maps between various types of almost contact metric manifolds. arXiv:2302.12677v1 [math.DG] (2023).
• Gherghe, C., Ianus, S., Pastore, A. M.: CR-manifolds, harmonic maps and stability. J. Geom. 71 (1-2), 42-53 (2001).
• Gherghe, C., Vîlcu, G.-E.: Harmonic maps on locally conformal almost cosymplectic manifolds. Commun. Contemp. Math., 26 (9), Article ID 2350052 (2024).
• Gorodski, C., Gusevskii, N.: Complete minimal hypersurfaces in complex hyperbolic space. Manuscripta Math. 103, 221-240 (2000).
• Gotoh, T.: The nullity of compact minimal real hypersurfaces in a complex projective space. Tokyo J. Math. 17 (1), 201-209 (1994).
• Gotoh, T.: The nullity of a compact minimal real hypersurface in a quaternion projective space. Geom. Dedicata 76 (1), 53-64 (1999).
• Gotoh, T.: The nullity of a compact minimal hypersurface in a compact symmetric space of rank one. Hokkaido Math. J. 33 (2), 429-441 (2004).
• Hsiang, W.-Y., Lawson, H. B.: Minimal submanifolds of low cohomogeneity. J. Differential Geom. 5, 1-38 (1971).
• Hu, Z., Yin, J., Li, Z.: Equivariant CR minimal immersions from S3 into CPn. Ann. Global Anal. Geom. 54 (1), 1-24 (2018).
• Ianus, S., Pastore, A. M.: Harmonic maps on contact metric manifolds. Ann. Math. Blaise Pascal 2 (2), 43-53 (1995).
• Ichiyama, T., Inoguchi, J., Urakawa, H.: Biharmonic map and bi-Yang-Mills fields. Note Mat. 28, suppl. 1, 233-275 (2009).
• Ichiyama, T., Inoguchi, J., Urakawa, H.: Classifications and isolation phenomena of biharmonic maps and bi-Yang-Mills fields. Note Mat. 30 (2), 15-48 (2010).
• Inoguchi, J.: On homogeneous contact 3-manifolds. Bull. Fac. Edu. Utsunomiya Univ. Sec. 2 59, 1-12 (2009).
• Inoguchi, J.: Harmonic maps in almost contact geometry. SUT J. Math. 50 (2), 353-382 (2014).
• Inoguchi, J., Sasahara, T.: Biharmonic hypersurfaces in Riemannian symmetric spaces I. Hiroshima Math. J. 46, 97–121 (2016).
• Inoguchi, J., Sasahara, T.: Biharmonic hypersurfaces in Riemannian symmetric spaces II. Hiroshima Math. J., 47 (3), 349-378 (2017).
• Iwata, K.: Classification of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits. Osaka J. Math. 15, 475-508 (1978).
• Jiang, G. Y.: 2-harmonic maps and their first and second variational formulas (in Chinese). Chinese Ann. Math. A 7, 389-402 (1986). English translation: Note Mat. 28, Suppl. 1, 209-232 (2009).
• Kacimi, B., Cherif, A. W.: Biharmonic submanifolds of quaternionic space forms. Kyungpook Math. J. 59 (4), 771-781 (2019).
• Kaup, W., Zaitsev, D.: On symmetric Cauchy–Riemann manifolds. Adv. Math. 149, 145-181 (2000).
• Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Amer. Math. Soc. 296, 137-149 (1986).
• Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurface in Pn(C). Math. Ann. 276 (3), 487-497 (1987).
• Kon, M.: Pseudo-Einstein real hypersurfaces in complex space forms. J. Differential Geom. 14 (3), 339-354 (1979).
• Li, Q.: Isometric immersions of generalized Berger spheres in S4(1) and CP2(4). J. Geom. Phys. 98, 21-27 (2016).
• Lichnerowicz, A.: Applications harmoniques et variétés kähleriennes. Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69), 341-402, Academic Press, London-New York, (1970).
• Maeda, S.: Hopf hypersurfaces with η-parallel Ricci tensors in a nonflat complex space form. Sci. Math. Japon. 76 (3), 449-456 (2014).
• Maeda, S., Udagawa, S.: Real hypersurfaces of a complex projective space in terms of holomorphic distribution. Tsukuba J. Math. 14 (1), 39-52 (1990).
• Martinez, A.: Ruled real hypersurfaces in quaternionic projective space. An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si, N. Ser., Sect. Ia 34 (1), 73-78 (1988).
• Mok, N., Siu, Y.-T., Yeung, S.-K.: Geometric superrigidity. Invent. Math. 113 (1), 57-83 (1993).
• Nagai, T., Kôjyô, H.: On some properties of hypersurfaces with certain contact structures. J. Fac. Sci. Hokkaido Univ. Ser. I 17 (1963), 160-167 (1963).
• Nagai, T., Kôjyô, H.: On some considerations of hypersurfaces in certain almost complex spaces. J. Fac. Sci. Hokkaido Univ. Ser. I 18, 114-123 (1964/65).
• Niebergall, R., Ryan, P. J.: Real hypersurfaces in complex space forms. Tight and Taut Submanifolds (T. E. Cecil and S. S. Chern, S.S. eds.). Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 233-305 (1997).
• Okumura, M.: Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphic sectional curvatures. Tôhoku Math. J. (2) 16, 270-284 (1964).
• Okumura, M.: Contact hypersurfaces in certain Kaehlerian manifolds. Tôhoku Math. J. (2) 18, 74-102 (1966).
• Okumura, M.: On some real hypersurfaces of a complex projective space. Trans. Amer. Math. Soc. 212, 355-364 (1975).
• Okumura, M.: Compact real hypersurfaces of a complex projective space. J. Differential Geom. 12, 595-598 (1977).
• Olszak, Z.: Curvature properties of quasi-Sasakian manifolds. Tensor, New Ser. 38, 19-28 (1982).
• Ou, Y.-L.: Biharmonic hypersurfaces in Riemannian manifolds. Pac. J. Math. 248 (1), 217-232 (2010).
• Ou, Y.-L., Chen, B.-Y.: Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry. Hackensack, NJ: World Scientific, (2020).
• Pérez-Barral, O.: Some problems on ruled hypersurfaces in nonflat complex space forms. Result. Math. 75 (4), Paper No. 167 (2020).
• Perrone, D.: Homogeneous contact Riemannian three-manifolds. Illinois J. Math. 13, 243-256 (1997).
• Perrone, D.: Hypercontact metric three-manifolds. C. R. Math. Acad. Sci. Soc. R. Can. 24 (3), 97-101 (2002).
• Perrone, D.: Remarks on Levi harmonicity of contact semi-Riemannian manifolds. J. Korean Math. Soc. 51 (5), 881-895 (2014).
• Petit, R.: Harmonic maps and strictly pseudoconvex CR manifolds. Commun. Anal. Geom. 10 (3), 575-610 (2002).
• Ren, Y., Yang, G.: Pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature. Calc. Var. Partial Differ. Equ. 57 (5), Paper No. 128 (2018).
• Sasahara, T.: Classification theorems for biharmonic real hypersurfaces in a complex projective space. Results Math. 74, Article number 13 (2019).
• Sasaki, S.: Spherical space forms with normal contact metric 3-structure. J. Differential Geom. 6, 307–315 (1971/72).
• Sharma, R.: Contact hypersurfaces of Kaehler manifolds. J. Geom. 78, 156-167 (2003).
• Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. (2) 88, 62-105 (1968).
• Siu, Y.-T.: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds. Ann. Math. (2) 112, 73-111 (1980).
• Suh, Y. J.: On real hypersurfaces of a complex space form with η-parallel Ricci tensor. Tsukuba J. Math. 14 (1), 27-37 (1990).
• Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495-506 (1973).
• Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures. J. Math. Soc. Japan 27, 43-53 (1975).
• Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures II. J. Math. Soc. Japan 27, 507-516 (1975).
• Takahashi, T.: Sasakian ϕ-symmetric spaces. Tôhoku Math. J. (2) 29, 91-113 (1977).
• Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space n complex variables. J. Math. Soc. Japan 14, 397-429 (1962).
• Tanaka, N.: A Differential Geometric Study on Strictly Pseudo-convex Manifolds. Kinokuniya Book-Store: Kinokuniya Book-Store, Tokyo, (1975).
• Tanno, S.: Some transformations on manifolds with almost contact and contact metric structures, I. Tôhoku Math. J. (2) 15 (2), 140-147 (1963).
• Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314 (1), 349-379 (1989).
• Tanno, S.: Pseudo-conformal invariants of type (1, 3) of CR manifolds. Hokkaido Math. J. 20 (2), 195-204 (1991).
• Urakawa, H.: CR rigidity of pseudo harmonic maps and pseudo biharmonic maps. Hokkaido Math. J. 46, 141-187 (2017).
• Vernon, M. H.: Contact hypersurfaces of a complex hyperbolic space. Tôhoku Math. J. (2) 39, 215-222 (1987).
• Wang, Q. M.: Real hypersurfaces with constant principal curvatures in complex projective spaces. I. Sci. Sin., Ser. A 26, 1017-1024 (1983).
• Wang, Z., Y.-L. Ou, Y.-L., Yang, H.: Biharmonic maps from tori into a 2-sphere. Chinese Ann. Math. Ser. B39 (5), 861-878 (2018).
• Webster, S. M.: Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13, 25-41 (1978).
• Yamaguchi, K.: Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations I. Nagoya Math. J. 62, 55-96 (1976).
• Yamaguchi, K.: Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations II. Nagoya Math. J. 69, 9-31 (1978).