Angles of holomorphy and the almost complex structure in a tangent bundle (Q2705840)

Any Riemannian manifold \((M,g)\) is naturally equipped with an almost Kähler structure on the tangent bundle \(TM\) defined by \textit{P. Dombrowski} [J. Reine Angew. Math. 210, 73-88 (1962; Zbl 0105.16002)]. NEWLINENEWLINENEWLINEThe author studies the angle of holomorphy of 2-planes in the tangent space of \(TM\). A formula is found which expresses the angle of holomorphy of a 2-plane \(Q\) as a function of three real numbers canonically associated with \(Q\), the verticality and the horizontality of \(Q\). The author also finds a formula involving the covariant derivative of the almost complex structure with respect to the Levi-Civita connection. NEWLINENEWLINENEWLINEWith these formulas, the author was able to produce examples of families of 2-planes along a curve on \(TM\) with constant angle of holomorphy, and examples of totally geodesic \(JP\)-submanifolds of the tangent bundle with the almost Kähler structure.