TITLE: Killing vector field of the sphere
QUESTION [7 upvotes]: Given a tangent vector field $X(x,y,z) = y\frac{\partial}{\partial x} -x\frac{\partial}{\partial y}$ of the sphere $S^2 \subset \mathbb{R}^3$.
Compute the Levi-Civita covariant derivative $\nabla_{v_p}X$ of any tangent vector $v_p$.
Secondly, show that this is a Killing vector field for the sphere.
I am having trouble with the first part, computing the covariant derivative.
Is the easiest way to compute it to use the ambient covariant derivative?

REPLY [4 votes]: for the first part
for an arbitary vector field $Y=y_1\dfrac{\partial}{\partial x} +y_2\dfrac{\partial}{\partial y}+y_3\dfrac{\partial}{\partial z}$ we have (at point p):
\begin{equation}
\nabla_{Y_p}X = y_1 (\dfrac{\partial}{\partial x} .y)\dfrac{\partial}{\partial x} -y_1(\dfrac{1}{\partial x}.x)\dfrac{1}{\partial y}+y_2 (\dfrac{\partial}{\partial y} .y)\dfrac{\partial}{\partial x} -y_2(\dfrac{\partial}{\partial y} .x)\dfrac{1}{\partial y}+y_3(\dfrac{\partial}{\partial z} .y)\dfrac{\partial}{\partial x} -y_3(\dfrac{\partial}{\partial z} .x)\dfrac{\partial}{\partial y}
\end{equation}
then
\begin{equation}
\nabla_{Y_p}X= y_2\dfrac{1}{\partial x}-y_1 \dfrac{1}{\partial y}
\end{equation}
for the second part, we have:
\begin{equation}
L_Xg(Y,X)=g(\nabla _Y X,Z)+g(\nabla _ZX,Y)=(y_2z_1-y_1z_2)+(z_2y_1-z_1y_2)=0
\end{equation}