| If P(--->)M is a principal H-bundle over a manifold M, then a Riemannian structure on M along with an Ad-invariant metric on the Lie algebra of H can be used to define the norm squared of the curvature of a connection in P. This is the Yang-Mills functional and the corresponding critical points, the so-called Yang-Mills connections, appear to play an important role in mathematics and physics.; We are primarily interested in the Yang-Mills functional for principal bundles of the form G(--->)G/H and for associated principal bundles; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); We assume that G, H, and U are compact Lie groups with G semisimple and we assume that the metric on G/H is defined by the negative of the Killing form on . These bundles over G/H have a canonical homogeneous (i.e., G-invariant) connection (omega)(,0) induced by the splitting = (CRPLUS) . Here is the orthogonal complement to in . We show that this connection is Yang-Mills and we study the stability properties of the functional near (omega)(,0).; By using Frobenius reciprocity and the correspondence between induced representations and sections of associated vector bundles, we can compute the second variation of the Yang-Mills functional at (omega)(,0). In particular, we show that the only compact irreducible Riemannian symmetric spaces which have nontrivial index or nullity at (omega)(,0) (for the bundle G(--->)G/H) are SO(p + 1)/SO(p) (p (GREATERTHEQ) 3); SP(p + 1)/SP(p) x SP(1); E(,6)/F(,4); F(,4)/Spin(9); and G x G/(DELTA) (G a simple Lie group). |
