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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?

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    Aren't you supposed to derive with respect to a vector *field*?2011-11-13
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    The exercise says a tangent vector, does that not make sense?2011-11-13
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    Can you prove that it either does or doesn't? Consider two vector fields $Y$ and $Z$ such that for some point $p \in S^2$ you have $Y|_p = Z|_p$. What can you say about $\nabla_Y X - \nabla_Z X$?2011-11-13
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    Actually, you're right, it does. Sorry, nevermind me then :)2011-11-13
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    In order to get some intuition on the geometrical meaning of Killing fields, it is usefull to know that the flow of a Killing field is by isometries. So geometrically it is kind of reasonable the field you have is Killing: it is exactly the field wich flow is by rotations around the $z$-axe of $\mathbb{R}^3$, so acts by isometries of the sphere. It is not a proof, but can help you to get some feeling.2012-04-22

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