In DoCarmo’s Riemannian Geometry, a vector field $X$ on a manifold $M$ is interpreted both as a map from $M$ to its tangent bundle $TM$, and when smooth as an operator on the set of smooth real-valued maps $\mathscr D$ of $M$.

After defining the bracket of two smooth vector fields $[X,Y]$, there comes the following theorem: If $X,Y$ are smooth vector fields on $M$ and $f,g$ are smooth real-valued maps on $M$, then $[fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X$.

My problem is, what does $fX$ mean?

  • 1
    $\begingroup$ It usually denotes pointwise product; $$(fX(g))|_p=f(p)X_p g, \qquad \forall p\in M, g\in C^\infty(M).$$ $\endgroup$ – Giuseppe Negro Nov 8 '18 at 13:14
  • $\begingroup$ Thank you I got it @GiuseppeNegro $\endgroup$ – user555729 Nov 8 '18 at 13:18

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