Hiroshima Mathematical Journal

On a Riemannian submanifold whose slice representation has no nonzero fixed points

Yuichiro Taketomi

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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.

Article information

Hiroshima Math. J., Volume 48, Number 1 (2018), 1-20.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

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


Taketomi, Yuichiro. On a Riemannian submanifold whose slice representation has no nonzero fixed points. Hiroshima Math. J. 48 (2018), no. 1, 1--20. doi:10.32917/hmj/1520478020. https://projecteuclid.org/euclid.hmj/1520478020

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