Tohoku Mathematical Journal
2021
March
SECOND SERIES VOL. 73, NO. 1

Tohoku Math. J.
73 (2021), 1-28

Title THE GAUSS MAPS OF TRANSVERSALLY COMPLEX SUBMANIFOLDS OF A QUATERNION PROJECTIVE SPACE

Author Kazumi Tsukada

(Received April 20, 2018, revised December 7, 2018)
Abstract. We study a kind of complex submanifolds in a quaternion projective space, which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. We treat them applying the theory of the quaternionic vector bundles. For a transversally complex immersion, we define a Gauss map whose values are complex structures of a quaternionic vector space. It is a generalization of “the mean curvature sphere” in the theory by Burstall, Ferus, Leschke, Pedit, and Pinkall. The Gauss map is a key notion for our theory. As an application, we show a characterization of complex projective spaces which are transversally complex submanifolds of a quaternion projective space.

Mathematics Subject Classification. Primary 53C26; Secondary 53B25.
Key words and phrases. A quaternion projective space, a quaternionic manifold, transversally complex submanifolds, Gauss maps.

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