A Geometric Interpretation of Cauchy-Schwarz Inequality in Terms of Casorati Curvature

Year 2018, Volume: 11 Issue: 1, 48 - 51, 30.04.2018

Nicholas D. Brubaker Bogdan D. Suceava

Abstract

In a visionary short paper published in 1855, Ossian Bonnet derived a theorem relating prescribed

curvature conditions to the admissible maximal length of geodesics on a surface. Bonnet’s work

opened the pathway for the quest of further connections between curvature conditions and

other geometric properties of surfaces, hypersurfaces or Riemannian manifolds. The classical

Myers’ Theorem in Riemannian geometry provides sufficient conditions for the compactness

of a Riemannian manifold in terms of Ricci curvature. In the present work, we are proving a

theorem involving sufficient conditions for a smooth hypersurface in Euclidean ambient space

to be convex, and the argument relies on an application of Cauchy-Schwarz inequality. This

statement represents, in consequence, a geometric interpretation of Cauchy-Schwarz inequality.

The curvature conditions are prescribed in terms of Casorati curvature.

Keywords

principal curvatures, Casorati curvature, smooth surfaces, smooth hypersurfaces, Cauchy-Schwarz inequality

References

• [1] Bonnet, Ossian: Sur quelque propriétés des lignes géodésiques, Comptes rendus de l’Academie des Sciences, 11 (1855), 1311–1313.
• [2] Calabi, Eugenio: On Ricci curvatures and geodesics, Duke Math. J., 34 (1967), 667–676.
• [3] Casorati, Felice, Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne, Acta Math. 14 (1) (1890), 95–110.
• [4] Chen, Bang-Yen, Geometry of submanifolds, M. Dekker, New York, 1973.
• [5] Chen, Bang-Yen, Geometry of submanifolds and its applications, Science University of Tokyo, 1981.
• [6] Chen, Bang-Yen, Pseudo-Riemannian submanifolds, -invariants and Applications, World Scientific, 2011.
• [7] doCarmo, Manfredo P., Riemannian Geometry, Birkhäuser, 1992.
• [8] Galloway, Gregory J., A Generalization of Myers Theorem and an application to relativistic cosmology, J. Diff. Geom., 14 (1979), 105–116.
• [9] Myers, S. B., Riemmannian manifolds with positive curvature, Duke Math. J., vol. 8 (1941), 401–404.