A Riemannian Geometric Invariant and Its Applications to a Problem of Borel and Serre

A new geometric invariant will be introduced, studied and determined on compact symmetric spaces. Introduction. We will introduce a new invariant on Riemannian manifolds, which is especially interesting on compact symmetric spaces, and we will determine the invariant for the compact symmetric spaces, thus amplifying the announcement [CN1]. A symmetric space M is defined with the point symmetry sx at every point x of M. Our new invariant, denoted by #jM, may be defined as the maximal possible cardinality #A2 of a subset A2 of M such that the point symmetry sx fixes every point of A2 for every x in A2. "The 2-number" #2Af is finite. #2M is clearly equal to 1 if M is not compact (but connected and simple). We thus consider compact spaces M only. When M is connected, the definition is equivalent to say that #2M is the maximal possible cardinality #A2 of a subset A2 of M such that for every pair of points, x and y, of A2 there exists a closed geodesic of M on which x and y are antipodal to each other. Thus the invariant could be defined on any connected Riemannian manifold. It is easy to see (1.4) that the geometric invariant #2M is a new obstruction to the existence of a totally geodesic embeddings /: N —> M, since the existence of / clearly implies the inequality #2./V < #2M. For example, while the complex Grassmann manifold G2(C4) of the 2-dimensional subspaces of the complex vector space C4 is obviously embedded into Gs(C6) as a totally submanifold, the space G2(C4)* which one obtains by identifying every member of G2(C4) with its orthogonal complement in C4, however, cannot be totally geodesically embedded into G3(C6)*, because #2G2(C4)* = 15 > 12 = #2G3(C6)* according to (6.4). The 2-number is not an obstruction to a topological embedding; for instance, the real projective space Gy(Rn) can be topologically embedded in a sufficiently high dimensional sphere, but the 2-number #2Gy(Rn) = n (> 2) simply prohibits a totally geodesic embedding into any sphere whose 2-number is 2 regardless of dimension. Nevertheless, the invariant, #2M, has certain bearings on the topology of M in other aspects; for instance, #2M equals xM, the Euler number of M, if Af is a semisimple hermitian symmetric space (4.3); (in particular, one thus has xM > X-B for every hermitian subspace B of a semisimple hermitian symmetric space M). And Received by the editors March 11, 1985 and, in revised form, May 8, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C21, 53C35, 22E40; Secondary 53C40.