Ngaiming Mok, HKU, Hong Kong On the projective
connection on Kähler manifolds of constant holomorphic sectional curvature Abstract: The complex projective space Pn,
the Euclidean space Cn
and the complex hyperbolic space form, equivalently the complex unit ball Bn,
share the common feature of possessing a complete Kähler
metric of constant holomorphic sectional
curvature. The Levi-Civita connections of
these canonical metrics agree with the natural holomorphic
projective connections. In particular, the second fundamental form on a
complex submanifold is holomorphic.
We examine applications of this elementary but basic fact which underlies the
geometries of Pn,
Cn
and Bn
and their quotient manifolds. An example is the proof, by means of
harmonic forms, that the tangent bundle of the ambient space form splits holomorphically on a compact complex submanifold
S if and only if S is totally geodesic, a well-known
result in the case of Pn
due to Van de Ven. (The Euclidean case was a recent
result established by Jahnke.) Another application is to the study of
holomorphic mappings between compact complex hyperbolic space forms. |