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Based on the E. Study’s map, a new approach describing instantaneous line congruence during the motion of the Darboux frame on a regular non-spherical and non-developable surface, whose parametric curves are lines of curvature, is proposed. Afterward, the pitch of general line congruence is developed and used for deriving necessary and sufficient condition for instantaneous line congruence to be normal. In terms of this, the derived line congruences and their differential geometric invariants were...

A new approach to the Ricci flow on $S^{2}$

J. Bartz, M. Struwe, R. Ye (1994)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

A New Approach to the Theory of the Riemann Space

Z. Janković (1984)

Zbornik Radova

A new characterization of Gromov hyperbolicity for negatively curved surfaces.

José M. Rodríguez, Eva Tourís (2006)

Publicacions Matemàtiques

In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces.

A new characterization of $r$ -stable hypersurfaces in space forms

H. F. de Lima, M. A. Velásquez (2011)

Archivum Mathematicum

In this paper we study the $r$ -stability of closed hypersurfaces with constant $r$ -th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$ -stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$ -th mean curvature.

A new characterization of the sphere in $R^{3}$

Thomas Hasanis (1980)

Annales Polonici Mathematici

Let M be a closed connected surface in $R^{3}$ with positive Gaussian curvature K and let $K_{I} I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I} I = c \sqrt H K^{r}$ , for some constants c and r, where H is the mean curvature of M.