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Connections with prescribed curvature

Dennis Deturck, Janet Talvacchia (1987)

Annales de l'institut Fourier

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 F , when does there exist locally a connection A such that F is the curvature of A  ? 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...

Constant Jacobi osculating rank of 𝐔 ( 3 ) / ( 𝐔 ( 1 ) × 𝐔 ( 1 ) × 𝐔 ( 1 ) )

Teresa Arias-Marco (2009)

Archivum Mathematicum

In this paper we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold M 6 = U ( 3 ) / ( U ( 1 ) × U ( 1 ) × U ( 1 ) ) . 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.

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