Anti-invariant Riemannian submersions from almost Hermitian manifolds. (English) Zbl 1207.53036

The concept of anti-invariant Riemannian submersion (anti-invariant R.s.) from an almost Hermitian manifold is introduced and studied.
More precisely, given an almost Hermitian manifold \((M,g,J)\) and a Riemannian manifold \((N,g')\), a Riemannian submersion \(F:M\to N\) is said to be anti-invariant if the vertical distribution ker\(F_*\) is anti-invariant with respect to \(J\). A Lagrangian Riemannian submersion \(F : M\to N\) is an anti-invariant R.s. such that \(J(\text{ker\,}F_*) = (\text{ker\,}F_*^\perp)\).
Firstly, the author gives an explicit example of Lagrangian R.s., Then, he studies the anti-invariant Riemannian submersions whose total space is a Kähler manifold. In this case, he states conditions that are equivalent to the integrability and the total geodesicity of the distribution \(\text{ker\,}F_*^\perp\), as well as to the total geodesicity of the vertical distribution. The author also characterizes the Lagrangian R.s. that are totally geodesic or harmonic. Finally, decomposition theorems for the total space of anti-invariant, in particular Lagrangian, Riemannian submersions are stated.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B20 Local Riemannian geometry
53C43 Differential geometric aspects of harmonic maps
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