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I was reading the literature about Riemannian submersions, and I came across the result showing the relation between the curvature tensor $\bar{R}$ in a manifold $M$ and the curvature tensor $R$ in a submersed manifold $N$.

Specifically, I am looking at Theorem 5 in Chapter 3, Section 5.2, page 83 of the book Peter Petersen, "Riemannian Geometry", 2nd ed, which states that

$$ g(R(X,Y)X,Y)=\bar{g}(\bar{R}(\bar{X},\bar{Y})\bar{X},\bar{Y})+\frac{3}{4} \left|[\bar{X},\bar{Y}]^V\right|^2 $$

where $g$ and $\bar{g}$ are the Riemannian metrics for $M$ and $N$, $X,Y$ are vector fields on $N$, $\bar{X},\bar{Y}$ are their horizontal lifts on $M$ and $\cdot^V$ denotes the projection of a field on the vertical bundle.

I can follow the entire proof that Petersen provides except for the very first step, where it says

We calculate the full curvature tensor, so let $X,Y,Z,H$ be vector fields on $N$ with zero Lie bracket. [in the book $N$ is $M$, but I think this is a typo]

The proof relies heavily on this assumption that all the brackets $[X,Y]$, $[X,Z]$, etc. are zero. My question is: why can we limit ourselves to this assumption? What happens in the general case where the brackets are not zero?

Sort-of-hint: In the original paper O'Neill, "The fundamental equations of a submersion", 1966, there is a similar result but without a detailed proof. In that paper it is suggested that we can make this assumption because $R$ is tensorial, and that the brackets can be made zero at least at a point. Unfortunately, I am not able to see why the latter is true, and how it can be applied.

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An important fact about tensors is that they depend only on the vector fields at a certain point, i.e. if $T$ is a tensor, $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ are vector fields, and $p$ is a point such that $X_i(p)=Y_i(P)$ for all $i$, then $T(X_1,\ldots,X_n)|_p=T(Y_1,\ldots,Y_n)|_p$.

In your example, we wish to evaluate the curvature of given vector fields at a point. An important fact about vector fields, is that given $v_1,\ldots,v_n\in T_pM$, there are vector fields $X_1,\ldots,X_n$ defined on some neighborhood, whose all brackets vanish at $p$, and for all $i\;X_i(p)=v_i$. This justifies Petersen's assumption.

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  • $\begingroup$ Oh, and if you want to understand the so called "important fact about vector fields", just take vector fields with constant coefficients with respect to some parametrization about $p$. $\endgroup$ – Amitai Yuval Aug 6 '14 at 21:46

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