Three-dimensional locally homogeneous Lorentzian affine hyperspheres with constant sectional curvature

Locally homogeneous affine hyperspheres with constant sectional curvature

“…Recall the following Lemma 2.2 (cf. Lemma 2.1 of [25]). If M is locally homogeneous a‰ne hypersurface, then for any p; q A M there exists a neighborhood U of p and A A SLðn þ 1; R nþ1 Þ y R nþ1 such that AðF ð pÞÞ ¼ F ðqÞ, AðF ðUÞÞ H F ðMÞ and A Ã ðxð pÞÞ ¼ xðqÞ.…”

“…We remark from [6,30] that a‰ne hyperspheres with constant sectional curvature and nonzero Pick invariant are homogeneous under unimodular a‰ne transformations. However, a‰ne hyperspheres with constant sectional curvature and vanishing Pick invariant are not necessarily homogeneous [25]. The solution of Problem 1 for dimension 2 and 3 has been obtained in [27] and [2] respectively, but there appear many implicit examples.…”

“…The classification of 3-dimensional case has been completed except for Lorentzian a‰ne hyperspheres (cf. [3,20,23,24,25]). For general dimension only partial results are known, see [4,5,15] for details.…”

“…cece li Theorem 1.3 (cf. Proposition 4.2 of [25]). Let M 3 be a locally homogeneous a‰ne hypersphere in R 4 with constant sectional curvature k and vanishing Pick invariant.…”

Four-dimensional locally strongly convex homogeneous affine hypersurfaces

“…Apart from the results of [14] and [9] most of these results deal with the case that the affine hypersurface is locally strongly convex, i.e. the induced affine metric is a positive definite metric.…”