Curvature and distance function from a manifold
type: Item
instance of: scholarly article
title: Curvature and distance function from a manifold
zbMATH Open document ID: 0941.53009
DOI: 10.1007/BF02922668
author: Luigi Ambrosio
author: Carlo Mantegazza
published in: The Journal of Geometric Analysis
publication date: 1998-09-02T00:00:00Z
review text: The paper is concerned with the relations between the differential invariants of a smooth manifold embedded in a Euclidean space \(\mathbb{R}^N\) and the square of the distance function from the manifold. In particular, we are interested in curvature invariants like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple way using third order derivatives of the squared distance function. Moreover, we study a general class of functionals depending on the derivatives up to a given order \(\gamma\) of the squared distance function and we find an algorithm for the computation of the Euler equation. Our class of functionals includes as particular cases the well-known area functional \((\gamma=2)\), the integral of the square of the quadratic norm of the second fundamental form \((\gamma=3)\) and the Willmore functional.
reviewed by: Luigi Ambrosio
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A55
zbMATH DE Number: 1186400
zbMATH Keywords: differential invariant
zbMATH Keywords: mean curvature
zbMATH Keywords: second fundamental form
zbMATH Keywords: squared distance function
zbMATH Keywords: Euler equation
zbMATH Keywords: area functional
zbMATH Keywords: Willmore functional
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1186400