An identity about Killing vector field
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I am studying Kähler geometry and I started reading the lecture note and this is the first exercise (p.2).
If $X$ is a Killing vector field on a Riemannian manifold and $Y,Z$ be two vector fields, then
begin{equation}
nabla^2X(Y,Z)+R(X,Y)Z=0
end{equation}
I thought by writing everything in local coordinates would work but I really want some kind of neat proof. Can someone give some suggestion?
Thank you in advance.
riemannian-geometry
asked Jan 8 at 19:44
Display NameDisplay Name
459313
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$begingroup$
I am studying Kähler geometry and I started reading the lecture note and this is the first exercise (p.2).
If $X$ is a Killing vector field on a Riemannian manifold and $Y,Z$ be two vector fields, then
begin{equation}
nabla^2X(Y,Z)+R(X,Y)Z=0
end{equation}
I thought by writing everything in local coordinates would work but I really want some kind of neat proof. Can someone give some suggestion?
Thank you in advance.
riemannian-geometry
asked Jan 8 at 19:44
Display NameDisplay Name
459313
$endgroup$
add a comment |
$begingroup$
I am studying Kähler geometry and I started reading the lecture note and this is the first exercise (p.2).
If $X$ is a Killing vector field on a Riemannian manifold and $Y,Z$ be two vector fields, then
begin{equation}
nabla^2X(Y,Z)+R(X,Y)Z=0
end{equation}
I thought by writing everything in local coordinates would work but I really want some kind of neat proof. Can someone give some suggestion?
Thank you in advance.
riemannian-geometry
asked Jan 8 at 19:44
Display NameDisplay Name
459313
$endgroup$
I am studying Kähler geometry and I started reading the lecture note and this is the first exercise (p.2).
If $X$ is a Killing vector field on a Riemannian manifold and $Y,Z$ be two vector fields, then
begin{equation}
nabla^2X(Y,Z)+R(X,Y)Z=0
end{equation}
I thought by writing everything in local coordinates would work but I really want some kind of neat proof. Can someone give some suggestion?
Thank you in advance.
riemannian-geometry
riemannian-geometry
asked Jan 8 at 19:44
Display NameDisplay Name
459313
asked Jan 8 at 19:44
Display NameDisplay Name
459313
asked Jan 8 at 19:44
Display NameDisplay Name
459313
asked Jan 8 at 19:44
Display NameDisplay Name
459313
asked Jan 8 at 19:44
Display NameDisplay Name
459313
459313
add a comment |
add a comment |
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