In each time t, let V=DxY is a vector field on M.
At time 0, d/dt( V ) = (Vt - V0)/t, this can be defined on each point of M.
<h, a> is the contraction of tensors. The result is a scalar. Just like the contraction between a vector and a co-vector:
<gradf, v>=<df, v> = v(f) = the directional derivative of f on the direction of v.