9
$\begingroup$

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of all non-positively curved metrics on a given manifold (broad curvature assumption), or maybe generic properties of metrics on a surface (broad dimension assumption). It is primarily for my own general knowledge, but may be it could help others as well to have a handy list lying around. As examples of what could be possible answers, let me begin the list with two generic properties that come to mind immediately:

  1. Generic simplicity of spectrum and generic properties of eigenfunctions: heuristically, the eigenvalues of the Laplacian for the generic metric on a compact manifold are non-repeated. Also, the eigenfunctions are generically Morse and have no nodal critical points (Uhlenbeck and Albert).

  2. The generic Riemannian metric on a smooth manifold is real analytic. This can either be realized by local approximation in charts, or by applying the Ricci flow and using a result of Bando (mr=888130).

Please add your own favorite generic property. Thanks!

Edit: As commented below, the word "generic" needs clarification. I am using it to refer to properties that are too strong for just any given metric to have, but if you perturb the metric slightly, you can produce a metric that now has the said property.

$\endgroup$
  • 5
    $\begingroup$ About number 2, are you also of the opinion that a generic function on $\mathbb{R}$ is real analytic? Such a position rather flies in the face of several notions of being "generic". $\endgroup$ – Igor Khavkine Oct 28 '15 at 15:13
  • 1
    $\begingroup$ I should rather be true that the generic metric on a Riemannian manifold is nowhere analytic. ;-) $\endgroup$ – Matthias Ludewig Oct 28 '15 at 17:10
  • 4
    $\begingroup$ I think you really mean "dense" rather than "generic". $\endgroup$ – Deane Yang Oct 28 '15 at 20:07
  • 1
    $\begingroup$ Does the perturbation have to be generic? That seems to be the difference between you meaning "dense" and meaning "generic" in the sense of the above comments. $\endgroup$ – Paul Reynolds Oct 28 '15 at 21:44
6
$\begingroup$

In a generic Riemannian manifold $M$ with dimension $\ge 3$ any convex set with smooth boundary is strictly convex; that is mid point for any two distint points in the set lie in the interior of the set. (In other words no geodesic of $M$ belongs to a smooth convex hypersurface $\Sigma\subset M$.)

| cite | improve this answer | |
$\endgroup$
6
$\begingroup$

I'm not sure if this is quite what you're looking for, but something that I have always found interesting is Wolpert's result that a generic compact Riemann surface is determined up to isometry by its length spectrum. The paper is here.

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ Yes, this certainly fits the bill. In fact, the proof given in Buser's book substantially shortens Wolpert's original proof by observing that a finite part of the marked length spectrum determines the entire spectrum. $\endgroup$ – Curios Oct 28 '15 at 22:29
5
$\begingroup$

The features of closed geodesics (periodic orbits of geodesic flows) associated to generic Riemannian metrics are described by the so-called "bumpy metric theorems": see e.g. page 2 of these papers here and here (or simply do a Google search with the keywords "bumpy metric theorem") for more details.

| cite | improve this answer | |
$\endgroup$
4
$\begingroup$

Here are two generic properties:

1) The restricted holonomy group of a generic Riemannian metric is $SO(n)$: http://mathworld.wolfram.com/HolonomyGroup.html.

2) The isometry group of the generic metric is trivial. That is to say, the only isometry is the identity. Even in the pseudo-Riemannian setting: http://arxiv.org/abs/1403.0182

| cite | improve this answer | |
$\endgroup$
3
$\begingroup$

Generic metric does not admit a lot of properties some special metrics admit. A good demonstration of this is the examples listed in the question (no multiple eigenvalues) or in the answers of
Matheus, Ben and Holonomia (and the property in the answer of Anton is of very different nature).

My item of the same kind is as follows:

(1) geodesic flow of the generic metric does not admit an integral which is polynomial or even analytic in momenta (see http://xxx.lanl.gov/abs/1510.01493 or Generic absence of non-trivial first integrals of geodesic flows). This is a local property.

(2) Morover, a geodesic flow of a generic metric on a closed manifold has a hyperbolic basic set which in particular implies that it has positive topological entropy, see http://annals.math.princeton.edu/2010/172-2/p01

Since there was a discussion in the comments to the question whether the author means ''generic'' or ''dense'', my examples work in both meanings.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.