The spheres in symmetric spaces (Q1175193)

A homomorphism between symmetric spaces \(M\) and \(N\) is a smooth map \(f: M\rightarrow N\) commuting with every point symmetry, \(fs_ x=s_{f(x)}f\). (Here \(s_ y\) denotes the point symmetry at \(y\) in \(M\) or \(N\).) When \(M\) is connected f is a homomorphism if and only if f is totally geodesic. In this paper the authors determine the totally geodesic spheres in every compact \textit{symmetric} space. As quoted by them, their method is based on a theory under development [see for example \textit{B. Y. Chen} and \textit{T. Nagano}, Duke Math. J. 45, 405-425 (1978; Zbl 0384.53024) and \textit{T. Nagano}, Tokyo J. Math. 11, No. 1, 57- 79 (1988; Zbl 0655.53041)]. An explanation of the geometric methods used, together with the basic definitions appear in Section 1. Roughly the method is the following. Given \(o\in S^ m\) and its antipodal, \(p\), they consider the polar \(M^ +(p)\), that is, the connected component of the fixed point set \(F(s_ o,M)\) containing p. If \(M^ -(p)\) denotes the connected component of \(F(s_ os_ p,M)\) through \(p\) then \(S^ m\subset M^ -(p)\); \(M^ - (p)\) is called the meridian. It was proved in \textit{Chen} and \textit{Nagano} [loc. cit.] that \(M^ +(p)\) and \(M^ -(p)\) are again symmetric spaces of complementary dimensions. Hence when the dimension of \(M^ +(p)\) is positive an induction argument can be used. When \(M^ +(p)=\{p\}\) they consider \(C(o,p)\), the centrosome, consisting of the midpoints of the geodesic segments joining \(o\) and \(p\). In this case (\(M^ +(p)=\{p\}\)) the sphere \(S^ m\) meets \(C(o,p)\) in the equator and the authors are able to reduce the question to a similar situation in a lower dimensional space. In Section 3 they establish the main theorem describing the spheres which are not contained in meridians and in Section 5 they show how to deal with the other spheres by using representation theory.