On slant immersions into Kähler manifolds

Abstract

Let φ: M-*N be an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with almost complex structure /. Then, φ is called slant if the angle between jφ*(X) and <p*(TpM) is constant for any X^TPM and any p^M. The typical examples of slant immersions are Kahler immersions and totally real immersions, where the slant angles are 0 and π/2, respectively. A slant immersion is called proper if it is neither a Kahler immersion nor a totally real immersion. In the case where M is a Riemann surface and N is a Kahler manifold, the slant angle was introduced as the Kdhler angle and studied by S. S. Chern and J. G. Wolf son [CW]. Examples of slant immersions of a Riemann sphere S into a complex projective space CP were given as the Veronese sequence of harmonic maps from S, which are classified in [BO] and [BJRW] in the case where S has constant curvature (see also [01]). The present concept of slant immersion was first introduced and studied by B. Y. Chen [C]. The examples of proper slant immersions into C are given in [C-T], Recently, Tazawa [T] has given examples of slant immersions into C with any given slant angle. However, there are a few results on slant submanifolds in CP. In this case, any general method to check whether given an immersion is slant or not is not known.