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On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Cícero P. Aquino, Henrique F. de Lima, Fábio R. dos Santos, Marco Antonio L. Velásquez (2015)

Commentationes Mathematicae Universitatis Carolinae

Our aim is to apply suitable generalized maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold, whose sectional curvature is supposed to obey standard constraints. In this setting, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

On Gauss-Bonnet curvatures.

Labbi, Mohammed-Larbi (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

On isometries of the symmetric space P₁(3,ℝ)

Gašper Zadnik (2014)

Colloquium Mathematicae

We classify the isometries in the non-identity component of the whole isometry group of the symmetric space of positive 3 × 3 matrices of determinant 1: we determine the translation lengths, minimal spaces and fixed points at infinity.

On nonisometric isospectral connected fractal domains.

Brian D. Sleeman, Chen Hua (2000)

Revista Matemática Iberoamericana

A fundamental question raised by M. Kac in 1966 is: Must two isospectral planar domains necessarily be isometric? Following a long history of investigation C. Gordon, D. L. Webb and S. Wolpert in 1992 finally proved that the answer is no. By using the idea of transposition maps one can construct a wide class of planar domains with piecewise continuous boundaries which are isospectral but non-isometric. In this note we study the Kac question in relation to domains with fractal boundaries and by following...

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