Totally umbilical real hypersurfaces in products of complex space forms without local product structure cannot have parallel shape operators, while those with local almost product structure are necessarily totally geodesic or extrinsic hyperspheres.
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On Umbilical Real Hypersufaces of Products of Complex Space Forms
Totally umbilical real hypersurfaces in products of complex space forms without local product structure cannot have parallel shape operators, while those with local almost product structure are necessarily totally geodesic or extrinsic hyperspheres.