I'm confused by a computation in Peter Petersen's Riemannian geometry book.
We consider $S^{2n+1}$ viewed as embedded in $\mathbb{C}^{n+1}.$ The circle $S^1$ acts naturally on $S^{2n+1}$ by complex multiplication. We let $V$ be the unit vector field tangent to this action. Then $\sqrt{-1}V$ is the inward unit normal vector field on $S^{2n+1} \subset C^{n+1}.$
Let $X,Y$ be vector fields on $S^{2n+1}.$ Petersen then claims:
$$g(Y, \sqrt{-1} \nabla_X^{\mathbb{C}^{n+1}} \sqrt{-1} V)=g(Y, \sqrt{-1}X),$$ where $g$ is the metric on $S^{2n+1}.$ Can anyone explain where this comes from?
[Note that I have omitted some information about this particular computation which seems irrelevant to my question.]