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Normal anti-invariant submanifolds of paraquaternionic Kähler manifolds. (English) Zbl 1139.53015

Paraquaternionic Kähler manifolds have been introduced by E. García-Río, Y. Matsushita and R. Vázquez-Lorenzo in [Rocky Mt. J. Math. 31, No. 1, 237–260 (2001; Zbl 0987.53020)]. In the paper under review, normal anti-invariant submanifolds of a paraquaternionic Kähler manifold are studied. The tangent bundle \(TN\) of such a submanifold \(N\) decomposes orthogonally into the direct sum \(TN=\mathcal D\oplus\mathcal D^{\perp}\), with \(\mathcal D\) being the paraquaternionic distribution on \(N\). The conditions for the distributions to be integrable are found. It turns out that if \(\mathcal D\) (resp., \(\mathcal D^{\perp}\)) is integrable, then the foliation generated by \(\mathcal D\) (resp., \(\mathcal D^{\perp}\)) is totally geodesic. Conditions for such a submanifold to be a semi-Riemannian product of leaves of the distributions are discussed.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C40 Global submanifolds

Citations:

Zbl 0987.53020
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