Isotropic affine spheres
The authors apply O'Neill's notion of isotropy to the study of affine hyperspheres contributing a piece to the grand open classification problem of affine hyperspheres. The details are as follows, where hyperspheres will be called spheres for short: In the equiaffine hypersurface theory of Berwald and Blaschke, one considers immersions \( f: M^{n} \to \mathbb R^{n+1} \) where the ambient space \( \mathbb R^{n+1} \) is only equipped with the standard volume element without regard of its Euclidean structure (Klein's geometry of the special affine group). Assuming strict local convexity, one can associate to \( f \) various differential-geometric objects like the affine normal vector field \( \xi \), the affine metric \( h \), and the induced symmetric connection \( \nabla \) on the parameter manifold \( M^{n} \). The metric \( h \) renders \( M^{n} \) a genuine Riemannian manifold (in case \( M^{n} \) is suitably oriented), and the Levi-Civita-connection \( \tilde{\nabla} \) of \( h \) gives rise to the difference tensor field \( K(X,Y) := \nabla_{X}Y-\tilde{\nabla}_{X}Y \), a relative to the classical Darboux tensor or cubic fundamental form. \( f \) is called an affine sphere if all normals go through a fixed point (proper case) or are mutually parallel (improper case). The proper spheres split in two subclasses: `elliptic' resp. `hyperbolic' ones, according to the position of the center on the concave resp. convex side of the sphere. While complete elliptic spheres are well known by the theorem of Blaschke and Deicke in the compact case and its complete extension (they are ellipsoids), and the asymptotic behaviour of complete hyperbolic spheres is clarified by the solution of the Calabi conjecture, it is still open how to classify in general the large variety of equiaffine spheres. See e.g. the book [\textit{A.-M. Li} et al., Global affine differential geometry of hypersurfaces. Berlin: Walter de Gruyter (1993; Zbl 0808.53002)], in particular Chapter 2. A bilinear vector-valued mapping \( B: V \times V \to W \), where \( V \) resp. \( W \) are real vector spaces equipped with non-degenerate scalar products \( g_{V} \) resp. \( g_{W} \)), is called TI{\( \lambda \)-isotropic} if \( g_{W}(B(v,v),B(v,v)) = \lambda g_{V}(v,v)^{2} \) for all \( v \in V \); see [\textit{B. O'Neill}, Can. J. Math. 17, 907--915 (1965; Zbl 0171.20503)]. Now, the present authors add to the sphere property the point-dependent isotropy of the difference tensor \( K \) (with \( g_{V} = g_{W} = h \) and non-vanishing factor) and show in dimensions \( n \geq 3 \) that there are only a few isolated copies sharing both properties, all described by canonical immersions of four specific symmetric spaces, namely: \( \mathbf{SL}(3,\mathbf{R})/\mathbf{SO}(3) \) (\( n = 3 \)), \( \mathbf{SL}(3,\mathbf{C})/\mathbf{SU}(3) \) (\( n = 8 \)), \( \mathbf{SU}^{\ast}(6)/\mathbf{Sp}(3) \) (\( n = 14 \)), \( \mathbf{E}_{6}(-26)/\mathbf{F}_{4} \) (\( n = 26 \)). The latter space rests on one of the exceptional Lie groups of É. Cartan's classification in a concrete description by matrices with octonions as entries. That it owes the required properties was discovered by the present authors and is in fact used in the large classification paper by \textit{H. Hu} et al. [J. Differ. Geom. 87, No. 2, 239--308 (2011; Zbl 1220.53015)] giving all the immersions with parallel Darboux tensor. There is a surprising relation to the normed division algebras. The second part of the proof then consist in identifying the general case with the examples by suitable frame choices and uniqueness results. The reviewer found these two works especially remarkable in depth and results.
scientific article; zbMATH DE number 6121159
scientific article