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Vector bundles structures on the acceleration bundle \(T^ 2 M\). (English) Zbl 0797.53025

Gheorghe, Atanasiu (ed.), The Proceedings of the sixth national seminar on Finsler, Lagrange and Hamilton spaces, Brasov, Romania, 1990. Brasov: Societatea de Stiinte Matematice din Romania Universitatea Transilvania Brasov, 129-136 (1990).
Let \(M\) be a smooth manifold and \(C_ x\) the set of smooth curves of \(M\) through \(x\) \(M\). Define on \(C_ x\) the equivalence relation: \(c \underset {x} \approx h\) iff \(\dot{c}(0) = \dot{h}(0)\) and \(\ddot{c}(0)= \ddot{h}(0)\), where \(\dot c\) and \(\ddot c\) denote the velocity and acceleration fields respectively of a curve \(c \in C_ x\). Then \(T^ 2_ x M = C_ x/_{\underset {x} \approx}\), thus obtaining the bundle \((T^ 2 M, M,p_ 2)\). The latter has a vector bundle (v.b) structure iff \(M\) admits a linear connection. In the present paper, the author considers the acceleration bundle \(l = (T^ 2 M,TM,p)\) and she proves that a section of 1 (i.e. a second order differential equation field) induces a v.b structrue on \(T^ 2 M\). A converse version of the previous result is also proved.
For the entire collection see [Zbl 0785.00031].

MSC:

53C05 Connections (general theory)
55R10 Fiber bundles in algebraic topology
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