Let $M$ be the total space of a fibre bundle with a base a compact simply connected manifold $X$. Using local coordinates and a partition of unity I was able to construct a riemannian metric $g_M$ such that $\pi:(M,g_M)\rightarrow (X,g_X)$ is a riemannian submersion. Then this riemannian metric $g_M$ will naturally define an horizontal distribution . However are we able to do the horizontal lift of a piecewise smooth path on $X$ which is uniquely defined provided we give the initial point ?
Any help or reference for this is appreciated. Thanks in advance.