**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 $R(X,Y) := [\nabla_X, \nabla_Y] - [X,Y]$ 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](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)