Tohoku Mathematical Journal
2024
June
SECOND SERIES VOL. 76, NO. 2

Tohoku Math. J.
76 (2024), 293-315

Title ON THE GEOMETRY OF A PAIR OF FOLIATIONS AND A CONFORMAL INVARIANT

Author Jerzy Kalina, Wojciech Kozłowski and Antoni Pierzchalski

(Received November 25, 2022, revised January 4, 2023)
Abstract. A pair of mutually orthogonal complementary foliations of a Riemanian manifold $(M, g)$ is considered. A geometry of such the pair is examined for a suitable torsion-free connection arising from the Bott connection. The connection has a number of nice geometric properties, however, it is not a metric one. The metrization leads to a connection with torsion. This metrized connection is a unique affine connection adapted to each of the foliations and with the torsion inducing selfadjoint endomorphisms of the tangent bundles (Theorem 1.1). Investigation of the geometry of the connection (Lemma 5.1) leads to a tensor which is a mixed measure of the “lack of the symmetry” of the Weingarten operator, so it encodes the extrinsic geometry of both foliations. This tensor is also a conformal invariant (Theorem 1.2).

Mathematics Subject Classification. Primary 53C12; Secondary 53B21, 53B05, 53A55.
Key words and phrases. Bott connection, orthogonal foliations, conformal invariant.

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