Abstract
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by do Carmo and Lima, as well as by do Carmo and Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.
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References
Alexander, S., Ghomi, M.: The convex hull property and topology of hypersurfaces with nonnegative curvature. Adv. Math. 180, 324–354 (2003)
Currier, R.J.: On hypersurfaces of hyperbolic space infinitesimally supported by horospheres. Trans. Am. Math. Soc. 313, 419–431 (1989)
Dajczer, M.: Submanifolds and isometric immersions. Based on the notes prepared by Maurício Antonucci, Gilvan Oliveira, Paulo Lima-Filho and Rui Tojeiro. Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX (1990)
Dajczer, M., Gromoll, D.: On spherical submanifolds with nullity. Proc. Am. Math. Soc. 93, 99–100 (1985)
do Carmo, M., Lima, E.: Isometric immersions with semi-definite second quadratic forms. Arch. Math. (Basel) 20, 173–175 (1969)
do Carmo, M., Warner, F.: Rigidity and convexity of hypersurfaces in spheres. J. Diff. Geom. 4, 133–144 (1970)
Ferus, D.: On the completeness of nullity foliations. Mich. Math. J. 18, 61–64 (1971)
Fomenko, V.T., Gajubov, G.N.: Unique determination of convex surfaces with boundary in Lobachevsky Space. Math. USSR, Sbornik No. 3 17, 373–379 (1972)
Guan, B., Spruck, J.: The existence of hypersurfaces of constant Gauss curvature with prescribed boundary. J. Diff. Geom. 62, 259–287 (2002)
Hadamard, J.: Sur certaines proprietés des trajectoires en dynamique. J. Math. Pures Appl. 3, 331–387 (1897)
Hadamard, J.: Les surfaces à courbure opposées et leurs lignes géodesique. J. Math. Pures Appl. 4, 27–73 (1898)
Jonker, L.: Immersions with semi-definite second fundamental forms. Can. J. Math. 27, 610–617 (1975)
Lawson, H.B.: The global behavior of minimal surfaces in \(S^n\). Ann. Math. 92(2), 224–237 (1970)
Obata, M.: The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature. J. Diff. Geom. 2, 217–223 (1968)
Osserman, R.: The convex hull property of immersed manifolds. J. Differential Geom. 6, 267–270 (1971/72)
Rodriguez, L., Tribuzy, R.: Reduction of codimension of regular immersions. Math. Z. 185, 321–331 (1984)
Sacksteder, R.: On hypersurfaces with no negative sectional curvatures. Am. J. Math. 82, 609–630 (1960)
Sacksteder, R.: The rigidity of hypersurfaces. J. Math. Mech. 11, 929–940 (1962)
Spivak, M.: A compreensive introduction to differential geometry—Vol IV, Publish or Perish (1979)
Stoker, J.: Über die Gestalt der positiv gekrümmten offenen Flächen im dreidimensionalen Raume. Compos. Math. 3, 55–88 (1936)
Acknowledgements
We would like to acknowledge Professor Manfredo do Carmo, who recently passed away, for his inspiring teaching and unwavering encouragement. His legacy will undoubtedly live on through us and many generations to come. The first author is also grateful to Marcos Dajczer, Luis Florit and Ruy Tojeiro for helpful conversations.
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In Memory of Manfredo do Carmo and Elon Lima.
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de Lima, R.F., de Andrade, R.L. Convexity, Rigidity, and Reduction of Codimension of Isometric Immersions into Space Forms. Bull Braz Math Soc, New Series 50, 119–136 (2019). https://doi.org/10.1007/s00574-018-0095-7
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DOI: https://doi.org/10.1007/s00574-018-0095-7