Geodesic
On the geometry of generalized almost quaternionic manifolds of vertical type
Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 33-44.

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We study generalized almost quaternionic manifolds of vertical type. Examples of this type of manifolds are given. It is proved that on a generalized almost quaternionic manifold there always exists an almost $\alpha$-quaternionic connection, which in the main bundle induces a metric connection. The criterion of the auto-duality of the projected vertical $2$-form on an almost $\alpha$-quaternion manifold is obtained. The components of the structural endomorphism on the space of the $G$-structure are obtained. The answer to the question is obtained: when does the Riemann-Christoffel endomorphism preserve the Kähler module of a variety. It is proved that the Riemann-Christoffel Hermitian endomorphism of an almost $\alpha$-quaternionic variety of vertical type preserves the Kähler module of a variety if and only if the structural sheaf of this variety is Einstein. Hence, as a consequence, we obtain that a four-dimensional manifold with a Riemannian or neutral pseudo-Riemannian metric is an Einstein manifold if and only if its module of auto-dual forms is invariant with respect to the Riemann-Christoffel endomorphism. The resulting corollary shows that the previous result is a broad generalization of the Atiyah-Hitchin-Singer theorem, which gives the Einstein criterion for 4-dimensional Riemannian manifolds in terms of auto-dual forms, since the result generalizes this theorem to the case of a neutral pseudo-Riemannian metric. On the other hand, this result is closely related to the well-known result of Berger, who clarifies it in the special case of quaternionic-Kähler manifolds: if a variety $M$ is quaternionic-Koehler, then its Riemann connectivity (and not just the Riemann-Christoffel operator) preserves the Koehler modulus of the variety. In this case, $M$ is an Einstein manifold.
Keywords: algebra of generalized quaternions, generalized almost quaternionic structure, quaternionic-Kaehler manifold, Einstein manifold. 
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O. E. Arsenyeva. On the geometry of generalized almost quaternionic manifolds of vertical type. Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 33-44. http://geodesic.mathdoc.fr/item/CHEB_2022_23_1_a3/

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