Biconservative hypersurfaces in space forms with harmonic curvature
DOI:
https://doi.org/10.2140/mscand.2026.132.273Keywords:
space form, mean curvature vector, biconservative hypersurface, harmonic curvature, Ricci curvatureAbstract
A biconservative hypersurface $M$ in a space form $\overline{M}^{n+1}(c)$ is a hypersurface whose normal part of the vector $\Delta\vec{H}$ vanishes. We study such hypersurfaces with harmonic curvature or parallel Ricci curvature, and prove that they have constant mean curvature. Also, by combining our results with previous results of H. B. Lawson Jr. and M. Umehara, we obtain that if $M$ is a biconservative hypersurface in a space form $\overline{M}^{n+1}(c)$ with parallel Ricci tensor or harmonic curvature, then the shape operator $\mathbf{A}$ of $M$ is covariant constant, i.e., $\nabla\mathbf{A}=0$.
References
A. L. Besse, Einstein manifolds, Ergebnisse der Math. (3) 10, Springer, 1987. https://doi.org/10.1007/978-3-540-74311-8
J.-P. Bourguignon, “Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein”, Invent. Math. 63:2 (1981), 263–286. https://doi.org/10.1007/BF01393878
R. Caddeo, S. Montaldo, C. Oniciuc, and P. Piu, “Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor”, Ann. Mat. Pura Appl. (4) 193:2 (2014), 529–550. https://doi.org/10.1007/s10231-012-0289-3
T. E. Cecil and P. J. Ryan, Geometry of hypersurfaces, Springer, 2015. https://doi.org/10.1007/978-1-4939-3246-7
B.-Y. Chen, Total mean curvature and submanifolds of finite type, 2nd ed., Series in Pure Mathematics 27, World Sci., Hackensack, NJ, 2015.
B.-Y. Chen, “Recent development in biconservative submanifolds”, pp. 229–285 in Analysis, algebra, and associated topics, edited by S. Singh et al., Springer, 2026. https://doi.org/10.1007/978-3-032-04418-1_19
Deepika, R. S. Gupta, and A. Sharfuddin, “Biharmonic hypersurfaces with constant scalar curvature in $E^5_s$”, Kyungpook Math. J. 56:1 (2016), 273–293. https://doi.org/10.5666/kmj.2016.56.1.273
A. Derdziński, “Classification of certain compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor”, Math. Z. 172:3 (1980), 273–280. https://doi.org/10.1007/BF01215090
A. Derdziński, “Compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor”, pp. 126–128 in Global differential geometry and global analysis (Berlin, 1979), edited by D. Ferus et al., Lecture Notes in Math. 838, Springer, 1981.
A. Derdziński, “Riemannian manifolds with harmonic curvature”, pp. 74–85 in Global differential geometry and global analysis (Berlin, 1984), edited by D. Ferus et al., Lecture Notes in Math. 1156, Springer, 1985. https://doi.org/10.1007/BFb0075087
D. DeTurck and H. Goldschmidt, “Regularity theorems in Riemannian geometry, II: Harmonic curvature and the Weyl tensor”, Forum Math. 1:4 (1989), 377–394. https://doi.org/10.1515/form.1989.1.377
J. Eells, Jr. and J. H. Sampson, “Harmonic mappings of Riemannian manifolds”, Amer. J. Math. 86 (1964), 109–160. https://doi.org/10.2307/2373037
D. Fetcu and C. Oniciuc, “Biharmonic and biconservative hypersurfaces in space forms”, pp. 65–90 in Differential geometry and global analysis—in honor of Tadashi Nagano, edited by B.-Y. Chen et al., Contemp. Math. 777, Amer. Math. Soc., Providence, RI, 2022. https://doi.org/10.1090/conm/777/15628
D. Fetcu, C. Oniciuc, and A. L. Pinheiro, “CMC biconservative surfaces in $S^n$ x $R$ and $H^n$ x $R$, J. Math. Anal. Appl. 425:1 (2015), 588–609. https://doi.org/10.1016/j.jmaa.2014.12.051
Y. Fu and M.-C. Hong, “Biharmonic hypersurfaces with constant scalar curvature in space forms”, Pacific J. Math. 294:2 (2018), 329–350. https://doi.org/10.2140/pjm.2018.294.329
A. Gray, “Einstein-like manifolds which are not Einstein”, Geom. Dedicata 7:3 (1978), 259–280. https://doi.org/10.1007/BF00151525
R. S. Gupta, “On bi-harmonic hypersurfaces in Euclidean space of arbitrary dimension”, Glasg. Math. J. 57:3 (2015), 633–642. https://doi.org/10.1017/S0017089514000524
R. S. Gupta, “Biconservative hypersurfaces in Euclidean 5-space”, Bull. Iranian Math. Soc. 45:4 (2019), 1117–1133. https://doi.org/10.1007/s41980-018-0188-5
R. S. Gupta and A. Arvanitoyeorgos, “Biconservative hypersurfaces in space forms $M^{n+1}(c)$”, Mediterr. J. Math. 19:6 (2022), art. id. 256. https://doi.org/10.1007/s00009-022-02134-y
T. Kashiwada, “On the curvature operator of the second kind”, Natur. Sci. Rep. Ochanomizu Univ. 44:2 (1993), 69–73.
K. Kendig, Elementary algebraic geometry, Graduate Texts in Mathematics 44, Springer, 1977.
U.-H. Ki, H. Nakagawa, and M. Umehara, “On complete hypersurfaces with harmonic curvature in a Riemannian manifold of constant curvature”, Tsukuba J. Math. 11:1 (1987), 61–76. https://doi.org/10.21099/tkbjm/1496160502
H. B. Lawson, Jr., “Local rigidity theorems for minimal hypersurfaces”, Ann. of Math. (2) 89 (1969), 187–197. https://doi.org/10.2307/1970816
I. Mogi and H. Nakagawa, “On hypersurfaces with parallel Ricci tensor in a Riemannian manifold of constant curvature”, pp. 267–279 in Differential geometry, edited by M. Obata et al., Kinokuniya, Tokyo, 1972.
S. Montaldo, “An invitation to biharmonic and biconservative submanifolds”, Note Mat. 45 (2025), 193–211. https://doi.org/10.1285/i15900932v45s1p193
S. Montaldo, C. Oniciuc, and A. Ratto, “Biconservative surfaces”, J. Geom. Anal. 26:1 (2016), 313–329. https://doi.org/10.1007/s12220-014-9551-9
E. Ōmachi, “Hypersurfaces with harmonic curvature in a space of constant curvature”, Kodai Math. J. 9:2 (1986), 170–174. https://doi.org/10.2996/kmj/1138037198
C. Oniciuc, “Biharmonic maps between Riemannian manifolds”, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 48:2 (2002), 237–248.
Y.-L. Ou and B.-Y. Chen, Biharmonic submanifolds and biharmonic maps in Riemannian geometry, World Sci., Hackensack, NJ, 2020.
P. J. Ryan, “Hypersurfaces with parallel Ricci tensor”, Osaka Math. J. 8 (1971), 251–259. http://projecteuclid.org/euclid.ojm/1200693241
S.-i. Tachibana, “A theorem on Riemannian manifolds of positive curvature operator”, Proc. Japan Acad. 50 (1974), 301–302. http://projecteuclid.org/euclid.pja/1195518988
N. C. Turgay and A. Upadhyay, “On biconservative hypersurfaces in 4-dimensional Riemannian space forms”, Math. Nachr. 292:4 (2019), 905–921. https://doi.org/10.1002/mana.201700328
M. Umehara, “Hypersurfaces with harmonic curvature”, Tsukuba J. Math. 10:1 (1986), 79–88. https://doi.org/10.21099/tkbjm/1496160391
