Generalized connections on $T(T^2M)$

Irena \v Comi\'c

Abstract: The geometry of some manifolds fibered over a given manifold $M$ is in the first place characterized by the group of allowable coordinate transformations. For the tangent manifold $TM$ these are given by $x^{i'}=x^{i'}(x)y^{i'} = \frac{\partial x^{i'}}{\partial x^{i}} y^{i}$, rank $\left[ \frac{\partial x^{i'}}{\partial x^{i}} \right] = n$, and for the total space of a vector bundle $E\rightarrow M$, we have $x^{i'} = x^{i'}(x)$, $y^{a'} = M^{a'}_{a}(x)y^{a}$, rank$(M^{a'}_{a}) = m =$ dimension of type fiber. \par In the last years R. Miron, Gh. Atanasiu and others examined the Osc$^{k}M$ spaces, [10], [11], [12]. Here the case $k=2$ will be investigated. Instead of Osc$^{2}M$ the notation $T^{2}M$ will be used (Osc$^{1}M$ coincides with $TM$). Instead of $d$-connection used in [10], [11], [12}], we consider here the generalized connection and determine its torsion tensor. As a special case the known $d$-connection is obtained.

Keywords: Second order connection, Torsion tensor

Classification (MSC2000): 53B25; 53B40

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