Abstract
For a Riemannian G-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost Hermitian and almost contact metric structures.
Highlights
Equipping an n-dimensional manifold M with a Riemannian metric g is equivalent to the reduction of a frame bundle L(M) to the orthogonal frame bundle O(M), i.e., to action of a structure group O(n)
We show that the divergence and integral formulas obtained in the previous section agree with the Walczak formulas [23]
Proposition 3 ([23]) On a closed Riemannian manifold equipped with a pair of complementary orthogonal and oriented distributions, the following Walczak integral formula holds smix = |H |2 + |H ⊥|2 + |T |2 + |T ⊥|2 − |B|2 − |B⊥|2, (13)
Summary
Equipping an n-dimensional manifold M with a Riemannian metric g is equivalent to the reduction of a frame bundle L(M) to the orthogonal frame bundle O(M), i.e., to action of a structure group O(n). This leads to the notion of an intrinsic torsion If this (1, 2)-tensor vanishes (in such case, we say that a G-structure is integrable), ∇ is a G-connection, which implies that the holonomy group is contained in G. One possible approach to curvature restrictions on compact G-structures can be achieved by obtaining integral formulas relating considered objects. This has been firstly done, in a general case, by Bor and Hernández Lamoneda [5]. For some Gray–Hervella classes, the characteristic vector field vanishes, and we get point-wise formula relating an intrinsic torsion to a curvature. We show, which is an immediate consequence of the formula for the Levi–Civita connection, that in these examples the characteristic vector field vanishes.
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