IJPAM: Volume 76, No. 3 (2012)

SOME DIFFERENTIAL GEOMETRIC INEQUALITIES FOR
SURFACES IN EUCLIDEAN SPACE WITH
NEGATIVE GAUSS CURVATURE

Serpil Karagöz$^1$, M. Kemal Sağel$^2$
Mathematics Department
Faculty of Art and Sciences
Abant Izzet Baysal University, TURKEY


Abstract. In this paper some differential geometric inequalities for surfaces in $E^{6}$ with negative Gauss curvature is derived. We compute some inequalities by means of the sum of betti numbers and Euler characteristics of surface.

  • $\displaystyle \int_{M^{4}}GdV\leq-\frac{4}{3}\pi^{2}\beta(M^{4})$,
  • $\displaystyle \int_{U}\frac{\sqrt[4]{-\lambda_{1}\lambda_{2}}}{\sqrt{\lambda_{1... ...rt{-\lambda_{2}}}[2\sqrt{-\lambda_{1}\lambda_{2}}+3(\lambda_{1}-\lambda_{2})]dV$
    $\displaystyle \geq\pi^{3}\beta(M^{4})+6\pi^{2}\chi(M^{4})-\frac{3}{2}\int_{U}\vert\alpha G(p)\vert dV$.


Received: February 29, 2012

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 3