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In Petersen's Riemannian geometry page 118 he computes the curvature operator on the product Riemannian manifold $S^n(1/a) \times S^m(1/b)$. For my question the fact that it is a product of spheres does not seem relevant - it could be any product manifold. In his computation he lets $Y$ be a unit vector field on $S^n$, $V$ a unit vector field on $S^n$ and $X$ be a unit vector field on either $S^n$ or $S^m$ that is perpendicular to $Y,V$. He then computes that $g(\nabla_YX,V) = 0$ via the Koszul formula. This all seems fine to me.

The part where I'm having trouble is he then claims that $g(\nabla_YX,V) = 0$ implies that $\nabla_YX = 0$ if $X$ is tangent to $S^m$ and $\nabla_YX$ is tangent to $S^n$ if $X$ is tangent to $S^n$. I'm not exactly sure how this result follows from $g(\nabla_YX,V) = 0$? It seems like maybe he is using the fact that the Levi-Cevita connection for a product manifold is essentially the sum of the respective connections when acting on vectors fields tangent to the manifolds in the product, but if so, why the discussion on $g(\nabla_YX,V) = 0$?

amc
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1 Answers1

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I had the same question before, and I think I just fugured it out: we have already showed $g(\nabla_YX,V) = 0$, use the same mathod, we can show $g(\nabla_VX,Y) = 0$. When $X$ is tangent to $S^m$, since $g(\nabla_YX,V) = 0$ for all unit vector field on $S^m$ (because $2g(\nabla_YX,X)$$ = \nabla_Y g(X,X)=0$), we know $\nabla_YX$ is tengent to $S^m$. Then we change the roles played by $X$ and $Y$, use $g(\nabla_VX,Y) = 0$ we can show $\nabla_XY$ is tengent to $S^n$. But since $[X,Y]=0$, $\nabla_XY=\nabla_YX$, so $\nabla_YX=0$.

Qhejaz
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