An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by . In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.
Osamu IKAWA. Takashi SAKAI. Hiroyuki TASAKI. "Weakly reflective submanifolds and austere submanifolds." J. Math. Soc. Japan 61 (2) 437 - 481, April, 2009. https://doi.org/10.2969/jmsj/06120437