Session 12. Geometry and Topology of Manifolds

Geometry of \(G\)-structures via intrinsic torsion

Kamil Niedziałomski, University of Łódź, Poland
We consider a \(G\)-structure on a manifold, i.e. the (oriented) Riemannian manifold such that the orthonormal frame bundle \(SO(M)\) has the reduction \(P\) of the structure group to \(G\subset SO(n)\). We additionaly assume that the quotient \(SO(n)/G\) is reductive. We study the geometry of \(P\) in \(SO(M)\). Mainly we consider extrinsic geometry. We show that minimality is equivalent to harmonicity of induced section of homogeneous bundle with respect to some modification of the Riemannian metric on the base manifold \(M\). This may lead to the slightly new concept of harmonic \(G\)--structure [1]. We give relevant examples [3]. Considerations are based on the study of properties of intrinsic torsion of \(G\)-structure, i.e. the section of adjoint bundle, which can be identified with the difference of the Levi-Civita connection and the \(G\)-connetion [2,4]
References
  1. J. C.Gonzalez-Davila, F. Martin Cabrera, Harmonic \(G\)--structures , Math. Proc. Cambridge Philos. Soc. 146, 2009, no. 2, 435--459.
  2. A. Gray, L. Hervella, The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants , Ann. Mat. Pura Appl. (4) 123, 1980, 35--58.
  3. K. Niedzia\l omski, On the frame bundle adapted to a submanifold , preprint, arXiv, http://arxiv.org/abs/1311.6172
  4. F. Tricerri, Localy homogeneous Riemannian manifolds , Rend. Sem. Mat. Univ. Poi. Torino Vol. 50, 1992, 411--426.
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