Fundamental vector field of a left action

For a smooth left action of a Lie group $G$ on $M$ , the fundamental vector field associated with $X\in\mathfrak g$ , in the positive-exponential convention, is

X_M(p)=\left.\frac{d}{dt}\right|_{t=0}\exp_G(tX)\cdot p.

Its flow is $p\mapsto\exp_G(tX)\cdot p$ , so this vector field is complete. The exponential specifies the sign as well as the direction.

Bracket sign

For this convention,

[X_M,Y_M]=-[X,Y]_M.

Indeed, on $G$ acting on itself by left multiplication these are right-invariant vector fields, whose bracket has the negative Lie-algebra sign. Orbit maps carry these related fields to the action on $M$ .

Consequently $X\mapsto-X_M$ preserves brackets. This is the sign that appears when differentiating inverse pullback on functions.

Comparison with a right action

For a right action, the analogous positive-exponential curve is $p\cdot\exp(tX)$ . That is the principal-bundle convention used elsewhere in this collection. One must specify the side of the action and the sign in the exponential before comparing formulas.