Connection Preserving Diffeomorphisms
The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (called affine transformation) as follows. The condition for the diffeomorphism $\Phi$ is that if $\Phi$ sends the vector fields $X$ and $Y$ to $\bar{X}$ and $\bar{Y}$, then $\nabla_{\bar{X}}\bar{Y} = \nabla_X Y$ (this holds for any vector fields $X$ and $Y$).
Infinitesimal Connection Preservation
They further investigate the infinitesimal condition for preserving a connection [KN §VI.2].
First condition [KN Prop. VI.2.5]
Define the torsion and curvature as usual by $T(X,Y) := \nabla_Y X - ∇_Y X - [X,Y]$, where $[\cdot,\cdot]$ is the Lie bracket of vector fields, and $R(X,Y) := [\nabla_X, \nabla_Y] - \nabla _{[X,Y]}$ where $[\cdot,\cdot]$ denotes the commutator and the Lie bracket respectively. Define the dual connection as $\bar{\nabla}_X Y := \nabla_X Y - T(X,Y)$. The condition on the vector field $X$ to be connection preserving is $$\nabla_V (\bar{∇}_W X) - \bar{∇}_{\nabla_V W} X = R(V, X) W$$
Second condition [KN Prop. VI.2.2]
Another condition is that the lift in the frame bundle of the vector field $X$ preserves the connection form.
Exponential Preserving Diffeomorphisms
They also observe [KN Prop. VI.1.1] that a connection preserving diffeomorphism $\Phi$ is also exponential preserving. This could be expressed informally by a commuting diagram as $\require{AMScd}$ $$ \begin{CD} TM @>T\Phi>> TM\\ @V \exp V V @VV \exp V\\ M @>>\Phi> M \end{CD}$$ This is informal since in general the exponential map $\exp$ associated to the connection $\nabla$ only sends a neighbourhood of the zero section of $TM$ to $M$, but the corresponding equation should hold as long at it makes sense.
Specific Question
What is the infinitesimal condition for a vector field to be exponential preserving?
(And how does that condition relate one of the above conditions on connection preservation, see my goal below)
Ultimate Goal
Understand exactly how much weaker exponential preservation is than connection preservation. For instance, with a flat connection (vanishing curvature and torsion), they are equivalent. When are these conditions not equivalent?
[KN]: Kobayashi & Nomizu, Foundations of Differential Geometry
