We give a partial local description of minimal hypersurfaces M 3 with identically zero Gauß-Kronecker curvature function in the unit 4-sphere S 4 (1), without assumption on the compactness of M 3 .

We consider minimal closed hypersurfaces M 4 ⊂ S 5 (1) with constant scalar curvature. We prove that, if M 4 is additionally a Willmore hypersurface, then it is isoparametric. This gives a positive answer to the question made by Chern about the pinching of the scalar curvature for closed minimal Willmore hypersurfaces in dimension 4.

We consider the difference tensor field of the Levi-Civita connections of the first and third fundamental form for non-degenerate hypersurface immersions in space forms. Within this framework, we characterize quadrics in terms of their principal curvature functions.

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