More on convolution of Riemannian manifolds
type: Item
instance of: scholarly article
title: More on convolution of Riemannian manifolds
zbMATH Open document ID: 1031.53099
published in: Beiträge zur Algebra und Geometrie
publication date: 2003-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/230982
full work available at URL: 2.html
review text: Let \((M_1,g_1)\) and \((M_2,g_2)\) be two Riemannian manifolds. Consider \(M_1\times M_2\) and the symmetric tensor field \(g_{f,h}\) of type \((0,2)\) on it defined by \[ g_{f,h}= h^2g_1+ f^2g_2+f h(df\otimes dh+dh \otimes df) \] for some positive differentiable functions \(f\) and \(h\) on \(M_1\) and \(M_2\), respectively. \(g_{f,h}\) (denoted by \(_hg_1* _fg_2)\) is called the convolution of \(g_1\) and \(g_2\) via \(h\) and \(f\). \(M_1\times M_2\) equipped with \(_hg_1* _fg_2\) is called a convolution manifold, denoted by \(_hM_1* _gM_2\). When this metric is nondegenerate, it defines a pseudo-Riemannian metric on \(M_1\times M_2\) with index \(\leq 1\) and then \(_hM_1* _fM_2\) is said to be a convolution pseudo-Riemannian manifold. When the index is zero, it is called a convolution Riemannian manifold. This kind of spaces and metrics have been introduced and studied by \textit{B. Y. Chen} in [Bull. Aust. Math. Soc. 66, 177-191 (2002; Zbl 1041.53012)] and the author continues this study in the paper under review. First he provides some new examples of convolution manifolds by using the tensor product of Euclidean submanifolds. Moreover, he derives the following necessary and sufficient condition for the convolution metric \(_hg_1* _fg_2\) of two Riemannian metrics to be a Riemannian metric: \(\|\text{grad} f\|_1 \|\text{grad} h\|_2 <1\) where \(\|\|\) denotes the length. The construction of the above mentioned examples leads the author to the construction of examples, and their study, of submanifolds in pseudo-Euclidean spaces whose distance function \(x\) satisfies \(\|\text{grad} x \|=c\in [0,\infty)\). Furthermore, he obtains a necessary and sufficient condition for one of the factors of a convolution Riemannian manifold to be totally geodesic and provides the complete list when the geodesic factor is an open interval of the real line. In the final section he completely classifies the flat convolution Riemannian surfaces.
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1894840
zbMATH Keywords: convolution manifold
zbMATH Keywords: convolution Riemannian manifold
zbMATH Keywords: convolution metric
zbMATH Keywords: conic submanifold
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: flat convolution Riemannian surface
zbMATH Keywords: tensor product immersion
author: Bang-Yen Chen
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1894840