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On the generalized Hamilton spaces. (English) Zbl 0804.53033
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 495-507 (1992).
The author considers generalized Hamilton spaces \(M^{* n}= (M,g^{ij}(x,p))\), where \(g^{ij}(x,p)\) is a symmetric \(d\)-tensor field on \(\widetilde{T^* M}= T^* M\to \{0\}\) (\(T^* M\) is the total space of the cotangent bundle of \(M\)) of type \((2,0)\) and of rank \(n\): \(\text{rank}\| g^{ij}(x,p)\|= n\) on \(\widetilde{T^* M}\). \(g^{ij}(x,p)\) is called the fundamental (or metric) \(d\)-tensor field of the space \(M^{* n}\).
If there exists a Hamiltonian \(H(x,p)\) such that \(g^{ij}(x,p)= {1\over 2} \dot\partial^ i\dot\partial^ j H\), then \(M^{* n}\) is reducible to a Hamilton space. When \(H(x,p)\) is 2-homogeneous with respect to \(p_ i\), \(M^{* n}\) is reducible to a Cartan space. A special class of generalized Hamilton spaces with the fundamental \(d\)-tensor field \(g^{ij}(x,p)= \gamma^{ij}(x,p)+ {1\over c^ 2} p^ i p^ j\) is also considered.
For the entire collection see [Zbl 0764.00002].
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)