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

Authors : Nicholas D. BRUBAKER, Bogdan D. SUCEAVA

Pages : 48-51

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Publication Date : 2018-04-30

Article Type : Research Paper

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