Spray (mathematics) - Correspondence With Nonlinear Connections

Correspondence With Nonlinear Connections

A semispray H on a smooth manifold M defines an Ehresmann-connection T(TM\0) = H(TM\0) ⊕ V(TM\0) on the slit tangent bundle through its horizontal and vertical projections

This connection on TM\0 always has a vanishing torsion tensor, which is defined as the Frölicher-Nijenhuis bracket T=. In more elementary terms the torsion can be defined as

Introducing the canonical vector field V on TM\0 and the adjoint structure Θ of the induced connection the horizontal part of the semispray can be written as hH=ΘV. The vertical part ε=vH of the semispray is known as the first spray invariant, and the semispray H itself decomposes into

The first spray invariant is related to the tension

of the induced non-linear connection through the ordinary differential equation

Therefore the first spray invariant ε (and hence the whole semi-spray H) can be recovered from the non-linear connection by

From this relation one also sees that the induced connection is homogeneous if and only if H is a full spray.