# Manifold with no Finsler structure?

Is there a good example for a smooth manifold to which one cannot give a Finsler structure in any meaningful way? Ideal the example should be of low dimension and not too bizarre.

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What does "meaningful" mean? Are Riemannian metrics constructed with partitions of unity not meaningful? –  Lee Mosher Mar 7 '12 at 4:55
Good point. I'm gonna say no, they may not be meaningful, since I vaguely remember that Yau once proved something that states you can modify a given Riemannian metric, under some constraints, and get a constant curvature. –  ssquidd Mar 7 '12 at 8:57
Without a description of what you consider to be "meaningful", this question is not answerable. –  Willie Wong Mar 7 '12 at 9:49
A finite-dimensional manifold admits a Finsler structure if and only if it admits as Riemannian structure (i.e. if and only if it is paracompact). –  alvarezpaiva Mar 7 '12 at 17:36
I see. Too bad I cannot delete my question now. –  ssquidd Mar 8 '12 at 5:23