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March 2018 On a Riemannian submanifold whose slice representation has no nonzero fixed points
Yuichiro Taketomi
Hiroshima Math. J. 48(1): 1-20 (March 2018). DOI: 10.32917/hmj/1520478020


In this paper, we define a new class of Riemannian submanifolds which we call arid submanifolds. A Riemannian submanifold is called an arid submanifold if no nonzero normal vectors are invariant under the full slice representation. We see that arid submanifolds are a generalization of weakly reflective submanifolds, and arid submanifolds are minimal submanifolds. We also introduce an application of arid submanifolds to the study of left-invariant metrics on Lie groups. We give a suffcient condition for a left-invariant metric on an arbitrary Lie group to be a Ricci soliton.


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Yuichiro Taketomi. "On a Riemannian submanifold whose slice representation has no nonzero fixed points." Hiroshima Math. J. 48 (1) 1 - 20, March 2018.


Received: 7 December 2016; Revised: 16 January 2017; Published: March 2018
First available in Project Euclid: 8 March 2018

zbMATH: 1345.53059
MathSciNet: MR3771997
Digital Object Identifier: 10.32917/hmj/1520478020

Primary: 53C25 , 53C40
Secondary: 53C30 , 53C44

Keywords: left-invariant metrics on Lie groups , minimal submanifolds , Ricci solitons , weakly reflective submanifolds

Rights: Copyright © 2018 Hiroshima University, Mathematics Program


Vol.48 • No. 1 • March 2018
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