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