Taiwanese Journal of Mathematics

NEW CURVATURE INEQUALITIES FOR HYPERSURFACES IN THE EUCLIDEAN AMBIENT SPACE

Charles T. R. Conley, Rebecca Etnyre, Brady Gardener, Lucy H. Odom, and Bogdan Suceavă

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Abstract

The spread of a matrix has been introduced by Mirsky in 1956. The classical theory provides an upper bound for the spread of the shape operator in terms of the second fundamental form of a hypersurface in the Euclidean space. In the present work, we are extending our understanding of the phenomenon by proving a lower bound, inspired from an idea developed recently by X.-Q. Chang. As we are exploring the very concept of curvature on hypersurfaces, we are introducing a new curvature invariant called amalgamatic curvature and we explore its geometric meaning by proving an inequality relating it to the absolute mean curvature of the hypersurface. In our study, a new class of geometric object is obtained: the absolutely umbilical hypersurfaces.

Article information

Source
Taiwanese J. Math., Volume 17, Number 3 (2013), 885-895.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705989

Digital Object Identifier
doi:10.11650/tjm.17.2013.2504

Mathematical Reviews number (MathSciNet)
MR3072267

Zentralblatt MATH identifier
1292.53008

Subjects
Primary: 53B25: Local submanifolds [See also 53C40] 53B20: Local Riemannian geometry 53A30: Conformal differential geometry

Keywords
principal curvatures shape operator extrinsic scalar curvature spread of shape operator surfaces of rotation absolutely umbilical hypersurfaces

Citation

Conley, Charles T. R.; Etnyre, Rebecca; Gardener, Brady; Odom, Lucy H.; Suceavă, Bogdan. NEW CURVATURE INEQUALITIES FOR HYPERSURFACES IN THE EUCLIDEAN AMBIENT SPACE. Taiwanese J. Math. 17 (2013), no. 3, 885--895. doi:10.11650/tjm.17.2013.2504. https://projecteuclid.org/euclid.twjm/1499705989


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