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I stumbled upon Problem 13-2 on p.344 in John Lee's Introduction to Smooth Manifolds (2nd Edition) where Lee explains that the proof for the existence of a Riemannian Metric on a manifold is done by a "Partition of Unity"-Argument and he emphasizes that a crucial part in the proof was that the set of inner products on a given tangent space is a convex subset of the vector space of all symmetric $2$-tensors.

Now it seems that the convexity property can not be omitted in "usual partition of unity" arguments but unfortunately, I don't understand why it's so important. Where exactly does the convexity of the (respective) subset comes into play if we want to do such partition of unity argument (in a much more general sense)?

So, to make my question a bit more precise:

My question: Say we want to patch together local objects to a global one (e.g. local sections of a vector bundle to a global section) by a usual partition of unity argument. Why and where do we need to use convexity of the subset containing the images of the objects we want to patch together? (in the case of local sections we require that there is an open set $V\subset E$ (total space) so that $V\cap E_p$ is a convex subset of the fiber $E_p$. Why do we need convexity of $V\cap E_p$ for example?

Thanks in advance for any help!

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    $\begingroup$ Just to add to the answer of @Kalejad: The problem is not to glue local objects to a global one, this can always be done. The question is whether you can deduce properties of the global object from properties of the local objects. In the case of symmetric two-tensors, this works for the property "is positive definite" but not for the property is non-degenerate. (For functions it would work for "is positive" but not for "is nowhere vanishing"). $\endgroup$ May 19 at 9:12
  • $\begingroup$ Thanks for the addtitional feedback, that's very helpful indeed! Thanks a lot. $\endgroup$
    – Zest
    May 19 at 15:08

1 Answer 1

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A convex subset of a vector space $C\subseteq V$ is a set which is closed under all convex combinations, which are wieghted sums whose wights are nonnegative and sum to $1$: $$ \lambda_1u_1+\lambda_2u_2+\cdots\lambda_nu_n \\ u_i\in U,\ \ \ \lambda_i\ge 0,\ \ \ \sum_i\lambda_i=1 $$ When combining together local sections $\sigma_i$ of vector bundles using a partition of unity $\psi_i$ in an expression like $\sum_i\psi_i\sigma_i$, we are effectively taking a convex combination at each fiber. This means that the combination retains any fiberwise properties which hold on a convex subset of each fiber (such as positive definiteness and symmetry of $(0,2)$ tensors).

The same argument doesn't work for indefinite metrics of arbitrary signature because these tensors do not form a convex subset, and so the wighted sum $\sum_i\psi_ig_i$ may fail to be a nondegenerate metric of the same signature..

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  • $\begingroup$ Hi Kajelad, thank you very much for your detailed answer. Sorry for the late reply. So is it correct to say that, very roughly speaking, we require the convexity in order to ensure that the smooth bump functions $\psi_i \in \mathcal C^\infty(U)$ we use in our partition of unity argument do indeed sum up exactly to $1$ ? (Here $U$ is just an open set of the base space) $\endgroup$
    – Zest
    May 19 at 15:10
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    $\begingroup$ @Zest No. A partition of unity $\psi_i$ satisfies $\psi_i\ge 0$, $\sum_i\psi_i=1$ by definition. When we combine local metrics with an expression like $g=\sum_i\psi_ig_i$, convexity is a condition on the $g_i$s, not the $\psi_i$s (i.e. "being a metric" is a property which holds on a convex subset of $T^0_2T_pM$). The reason we care about convexity is because the sum $\sum_i\psi_ig_i$ is a convex combination of the $g_i$s. $\endgroup$
    – Kajelad
    May 19 at 22:06
  • $\begingroup$ Hi Kajelad, i see! That clarifies it. Thank you very much, your help is highly appreciated. $\endgroup$
    – Zest
    May 19 at 22:29
  • $\begingroup$ Sorry, there is one remaining question. What would be the property for the case of smooth local sections that we would like to have preserved that requires to make use of the convexity argument? $\endgroup$
    – Zest
    May 20 at 0:31
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    $\begingroup$ @Zest You can always combine local sections together in this manner to obtain a well-defined global section. It's just that if these local sections have some additional local property (such as being a volume form/affine connection/Riemannian metric) the resulting global section will only be assured to have that property if that property is convex. $\endgroup$
    – Kajelad
    May 20 at 2:11

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