What does one mean when one says that a certain manifold 'admits' a certain metric?
Not sure if this is the appropriate context but some examples:
Every manifold admits a Riemannian metric. This simply means that there are Riemannian metrics on the manifold.
The 4-sphere does not admit a Lorentzian metric. This means there is no Lorentzian metric on the 4-sphere.
Hopefully that helps some. This is the typical usage I am most familiar with, but my interests are also pretty far from differential geometry, so it would not be surprising if there are other common usages I am simply unaware of.
You may remember that a metric induces a topology on a space in a natural way (take as open sets those sets that are open under the metric). Strictly speaking, you can define whatever metric you want on a manifold, but you generally want the metric to be compatible with the underlying topology (and smooth structure) in some way. Take the reals with a non-Hausdorff topology; clearly the Euclidean metric is completely incompatible here, since it naturally gives rise to a Hausdorff topology. That's not to say that you can't put a Euclidean metric on your space, but it won't really have any connection with the topology (in fact, the metric will naturally give rise to a completely different (i.e. Hausdorff, and so non-homeomorphic) topology).
To add a bit more intuition for this:
A "manifold" is a topological object and does not have a pre-defined geometrical shape. For example, a sphere ##S^2## can be the usual round sphere we know, but it can also be an ellipsoid, or a wobbly shape, or any smooth 2-dimensional object that closes in on itself, is orientable, and has no handles.
To put a metric on a manifold means to give it a well-defined shape*. So, if we put the metric
[tex]ds^2 = d\theta^2 + \sin^2 \theta \; d\phi^2[/tex]
on the sphere, then it becomes the usual round sphere we know. But we could also put a metric on it to make it an ellipsoid, etc. The sphere admits many different metrics.
However, there are some obstructions that come from the sphere's topology. The sphere does not admit a flat metric, and it does not admit a metric with too much negative curvature. The total curvature integrated over the whole sphere must be positive.
* Footnote on "shape": Be careful about imagining the "shape" of a manifold, because a lot of our visual intuitions about the "shape" of things actually depend on some embedding of the object into ##R^3##. But in intrinsic geometry, we are only concerned with the aspects of the "shape" of something that do not depend on the embedding. For example, if I take a flat piece of paper and bend it slightly, it is still flat from the intrinsic point of view.
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