Weakly reflective orbits and tangentially degenerate orbits of $s$-representations


Takashi Sakai 

16:10 - 17:00 

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

This talk is based on a joint work with Osamu Ikawa and Hiroyuki Tasaki. A linear isotropy representation of a Riemannian symmetric space is called an $s$-representation. In this talk we will study some geometric properties of orbits of $s$-representations as submanifolds in the Euclidean sphere. We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we give the classification of weakly reflective orbits and austere orbits of $s$-representations. We observe that some wealky reflective orbits have a remarkable property, namely, they are tangentially degenerate. Furthermore these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.