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Topology of quadratic maps and Hessians of smooth maps. (English. Russian original) Zbl 0719.58006
J. Sov. Math. 49, No. 3, 990-1013 (1990); translation from Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 26, 85-124 (1988).
Let K be a cone in \({\mathbb{R}}^ k\) and \(K^*=\{\omega \in ({\mathbb{R}}^ k)^*:\omega\) (x)\(\leq 0\) for all \(x\in K\}\) its dual cone. Consider a symmetric bilinear map p on \({\mathbb{R}}^{N+1}\) with values in \({\mathbb{R}}^ k\). For \(\omega \in K^*\setminus \{0\}\) denote by \(\omega\) P the operator on \({\mathbb{R}}^{N+1}\) satisfying \(\omega p(x,y)=((\omega P)x,y)\) for \(x,y\in {\mathbb{R}}^{N+1}\) and assume that for \(\omega \in K^*\setminus \{0\}\) we have that (\(\omega\) P)x\(\neq 0\) whenever \(x\in {\mathbb{R}}^{N+1}\setminus \{0\}\) is such that p(x,x)\(\in K\). Call \(S^{k- 1}\) and \(S^ N\) the unit spheres in \(({\mathbb{R}}^ k)^*\) and \({\mathbb{R}}^{N+1}\), let \(\Omega:=K^*\cap S^{k-1}\) and \(B:=\{(\omega,x)\in \Omega \times S^ N:\omega p(x,x)>0\}\) and denote by \(\beta_ 1: B\to \Omega\) the projection onto the first factor. For \(\omega\in \Omega\), order the eigenvalues \(\lambda_ n(\omega P)\) of \(\omega\) P in increasing order and let \(\Omega_ n=\{\omega \in \Omega:\lambda_{n+1}(\omega P)\geq 0\}.\)
The author proves that for an abelian group \({\mathcal A}\) we have that \(H^ i(\Omega_{N-j},\Omega_{N-j-1};{\mathcal A})=E^{ij}({\mathcal A})\), where \((E_ r({\mathcal A}),d_ r)\) is the Leray spectral sequence for \(\beta_ 1\). Moreover, for \({\mathcal A}={\mathbb{Z}}_ 2\), he computes the differential \(d_ 2\) in terms of cup products. Similar results are obtained when \({\mathbb{R}}^{N+1}\) is replaced by a separable Hilbert space. Finally, these results are applied to second derivatives of differentiable mappings.

58D15 Manifolds of mappings
58C25 Differentiable maps on manifolds
Full Text: DOI
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