Intrinsic and extrinsic geometry of ovaloids and rigidity (Q2735681)

The authors prove the following rigidity result: Let \(x, x^\#:M \rightarrow E^3\) be two ovaloids with nowhere dense umbilics and such that at any \(p \in M\) \(S(p) = S^\#(p)\) and \(\omega (p) = \omega ^\#(p)\), where \(S, S^\#\) are the respective Weingarten operators and \(\omega, \omega ^\#\) are the respective volume form of spherical images. Then \(x\) and \(x^\#\) are congruent. NEWLINENEWLINENEWLINEThe authors also show that in case when \(x\) is an ovaloid of revolution the assumption regarding the volume forms of the spherical image can be omitted and the conclusion is still true.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].