Kaehler Immersions With Trivial Normal Connection

Ins note we shall prove that any complex submanifold of aomplex spaco fbm(ie Kaehler manifold of constant hol6morphic sectio na1 curvatUre)withe trivial noimaloection is totally geodesic This is the extension of Theorem 70f Nonu-Smyth3f()r complex hypersurfaces Of a complex spafbrm For a complex hypersurfaoe which is Ehlstein Smyth4has obtaid the classCation of it and Chem1has proved the corresponding local theorem If the normali onnection of a complex hypersurface of a eomplex space f()rm is trivial then it is Einstein alld the holomohic sectional curvature of the anibient manifold is smaller than zero or equal to zero alld colhbing the Theorem of Chern1 it is totally geodesic3 This resultue fbr any codimension Our theorem states that thei Kaehler immersion into a complex space fbrm with trivial normal copnection Qccur only When the ambient maQld and submanld qrQ 1)oth fiat Kaehler manifolds 1Prelimioari Let4 be a Kaehler manifold of cQmplex dimension r The Kaehler structure and the Kaehler metric oflf wbe denoted by J and resptively LtM be a COmplex(mensional manifold and1)e a complex