ON EULER'S THEOREM IN SEMI-EUCLIDEAN SPACES $\mathbb{E}_{v}^{n+1}$

Abstract

In this paper, we study Euler's theorem for semi-Euclidean hypersurfaces in the semi-Euclidean spaces [Formula: see text]. We obtain an analog of the well-known Euler's theorem for semi-Euclidean hypersurfaces in the semi-Euclidean spaces [Formula: see text]. Then we give corollaries of Euler's theorem concerning conjugate and asymptotic directions. After that, we express Euler's theorem and its corollaries for hypersurfaces in the Euclidean space 𝔼m in the case n = m - 1, v = 0. In addition, we give the well-known Euler's theorem and its corollaries for surfaces in the case n = 2, v = 0, for Lorentz surfaces in the case n = 2, v = 1 and for hypersurfaces in Lorentz spaces in the case n = m - 1, v = 1.