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The spray and antispray theory in the subspaces of Miron’s \(Osc^kM\). (English) Zbl 1224.53040

Let \(E=Osc^kM\) be a \((k+1)n\)-dimensional \(C^\infty\) manifold, where \(M\) is a basic manifold of class \(C^\infty\) and dimension \(n\), such that every point \(u\in E\) in a local chart \((V,\psi)\) has the coordinates \((y^{0a},y^{1a},\dots,y^{ka})\), \(A=0,1,2,\dots,k\), \(a=1,2,\dots,n\), where \(y^{0a}\in M\) is the point of some curve \(c(t):\mathbb{R}\to M\), and the coordinates are connected by the relation \(y^{Aa}=\frac{d}{dt}y^{(A-1)a}\), \(A=1,2,\dots k\). The authors consider specially adapted bases of \(T(E)\) and the \(k\)-tangent structure \(J\) to study a \(k\)-spray on \(T(E)\) and its integral curve as well as a \(k\)-antispray on \(T^\star(E)\) and its integral curve. They also consider the spray and antispray theories in the subspaces of \(E\).

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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