Revisiting Liebmann’s theorem in higher codimension

• Autorzy: Araújo, Jogli G. , Lima, Henrique F. de
• Języki publikacji: EN

Abstrakty

EN

We deal with compact surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in a space form Q_c^2+p of constant sectional curvature c ϵ {−1, 0, 1}. Such a surface is called an LW-surface when it satisfies a linear Weingarten condition of the type K = aH + b for some real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively. In this setting, we extend the classical rigidity theorem of Liebmann (1899) showing that a non-flat LW-surface with non-negative Gaussian curvature must be isometric to a totally umbilical round sphere.

Słowa kluczowe

EN

space forms; compact LW-surfaces; parallel normalized mean curvature vector field; totally umbilical round spheres

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2019
• Tom: Vol. 67, no. 2
• Strony: 179--185
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Araújo, Jogli G.

• Departamento de Matemática, Universidade Federal Rural de Pernambuco, 52.171-900 Recife, Pernambuco, Brazil,

autor

Lima, Henrique F. de

• Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil,

Bibliografia

• [1] J. A. Aledo, L. J. Alías and A. Romero, A new proof of Liebmann classical rigidity theorem for surfaces in space forms, Rocky Mountain J. Math. 35, (2005), 1811-1824.
• [2] M. Dajczer, Submanifolds and Isometric Immersions, Math. Lect. Ser. 13, Publish or Perish, Houston, TX, 1990.
• [3] J. Hadamard, Sur certaines propriétés des trajectoires en dynamique, J. Math. Pures Appl. 3 (1897), 331-387.
• [4] D. Koutroufiotis, Two characteristic properties of the sphere, Proc. Amer. Math. Soc. 44 (1974), 176-178.
• [5] H. Liebmann, Eine neue Eigenschaft der Kugel, Nachr. Königl. Ges. Wiss. Göttingen Math.-Phys. Kl. 1899, 44-55.
• [6] S. Montiel and A. Ros, Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, in: Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. 52, Longman Sci. Tech., Harlow, 1991, 279-296.
• [7] A. Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamer. 3 (1987), 447-453.
• [8] A. Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27 (1988), 215-220.
• [9] R. Schneider, Closed convex hypersurfaces with second fundamental form of constant curvature, Proc. Amer. Math. Soc. 35 (1972), 230-233.
• [10] U. Simon, Characterizations of the sphere by the curvature of the second fundamental form, Proc. Amer. Math. Soc. 55 (1976), 382-384.
• [11] J. Weingarten, Ueber eine Klasse auf einander abwickelbarer Flächen, J. Reine Angew. Math. 59 (1861), 382-393.
• [12] J. Weingarten, Ueber die Flächen, derer Normalen eine gegebene Fläche berühren, J. Reine Angew. Math. 62 (1863), 61-63.
• [13] D. Yang and Z. Hou, Linear Weingarten spacelike submanifolds in de Sitter space, J. Geom. 103 (2012), 177-190.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).

Identyfikatory

DOI

10.4064/ba190514-30-5