Connections on embedded manifolds.
We discuss the problem of prescribing the curvature of a connection on a principal bundle whose base manifold is three-dimensional. In particular, we consider the local question: Given a curvature form , when does there exist locally a connection such that is the curvature of ? When the structure group of the bundle is semisimple, a finite number of nonlinear identities arise as necessary conditions for local solvability of the curvature equation. We conjecture that these conditions are also...
In this paper we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold . As a consequence, an explicit expression of the Jacobi operator independent of the geodesic can be obtained on such a manifold. Moreover, we show the way to calculate the Jacobi vector fields on this manifold by a new formula valid on every g.o. space.