Does this property hold for fundamental vector fields?

Question

Let G be a lie group and let M be a G-manifold. Let $\mathfrak{g}$ be the lie algebra of G. If $X \in \mathfrak{g}$, we denote by $X_M$ the vector field on M such that

$(X_M.f)(m) = \frac{d}{dt} f (exp(-tX)m)|_{t=0}$

for $f \in C^\infty(M), m \in M.$

Suppose that $X = Y + Y'$ , where $Y,Y' \in \mathfrak{g}$, $\lambda \in \mathbb{R}$. Does this imply that $X_M = Y_M + Y'_M$ ?

Answer 1

Let $\phi:G\rightarrow M$ defined by $\phi(g)=g.m$, $f(exp(-tY).m)=f(\phi(exp(-tY))$,

${d\over{dt}}f(exp(-tY)m)_{\bar t=0}={d\over {dt}}_{t=0}f(\phi(exp(-tY))=df_m.d\phi_{Id}.{d\over {dt}}_{t=0}exp(-tY))={d\over {dt}}_{t=0}f(\phi(exp(-tY))=df_m.d\phi_{Id}(-Y)$.

This implies that $(Y_M+Y'_M)(m)=df_m.d\phi_{Id}(-Y)+df_m.d\phi_{Id}(-Y')=df_m.d\phi_{Id}(-(Y+Y'))=X_M(m)$.