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Improper affine spheres and \(\delta\)-invariants. (English) Zbl 1089.53011

Opozda, Barbara (ed.) et al., PDEs, submanifolds and affine differential geometry. Proceedings of the conference and autumn school, Bȩdlewo, Poland, September 23–27, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 69, 157-162 (2005).
The Calabi composition \(M\) of the improper affine spheres \(M_1\) and \(M_2\) with graph representation \(x_{p+1}= F_1(x_1,\dots, x_p)\) resp. \(y_{q+1}= F_2(y_1,\dots, y_q)\), corresponding to affine normal vectors \((\undersetbrace p\to{0,\dots,0,}1)\) resp. \((\undersetbrace q\to{0,\dots,0},1)\), is defined by \(z= F_1(x_1,\dots, x_p)+ F_2(y_1,\dots,y_q)\) with affine normal vector \((\undersetbrace{p+q}\to{0,\dots,0},1)\); it is again an improper affine sphere. The authors introduce an inequality for some deficiency invariant \(\delta^{\sharp}(n_1,\dots, n_k)\) for certain scalar curvature entities of the curvature tensor \(\widehat R\), belonging to the affine fundamental form \(h\) of a graph hypersurface \(M\) in \(\mathbb{R}^{n+1}\) where \(h\) is supposed to be positive definite. Here \((n_1,\dots, n_k)\) is a \(k\)-tuple of integers \(\geq 2\) satifying \(n_1< n\) and \(n_1+\cdots+ n_k\leq n\) \((k\geq 0)\). Their main result says that if \(\dim(\text{Im\,}K)= n\) (\(K=\) difference tensor of the induced connection \(\nabla\) of \(M\) and the Levi-Cività connection \(\widehat\nabla\) of \(h\)) and if equality holds in the inequality for \(\delta^{\sharp}(n_1,\dots, n_k)\) then \(M\) is locally the Calabi composition of \(k\) improper affine spheres of dimensions \(n_1,\dots,n_k\).
For the entire collection see [Zbl 1077.53002].

MSC:

53A15 Affine differential geometry
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