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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].
##### MSC:
 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)