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redrzewski
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I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric.
thanks
thanks
zhentil said:The proof that every manifold has a metric (as well as the proof of Whitney's embedding theorem) relies on paracompactness. If you drop this requirement, you can have all sorts of aberrations.
In fact, if your space has a metric, it has to be second countable (delta-balls type argument).
True, true. The space must be path-connected for what I said to hold.hamster143 said:We have to be careful with definitions here. In particular, it seems to me that a non-second-countable smooth manifold may be equipped with a local Riemann metric but not be a metric space.
zhentil said:True, true. The space must be path-connected for what I said to hold.
A differentiable manifold that is not Riemannian is a type of mathematical structure that allows for smooth, continuous functions to be defined on it, but does not necessarily have a metric tensor that satisfies the laws of Riemannian geometry. This means that the concept of distance and angles may not be well-defined on the manifold, making it a more abstract object.
A Riemannian manifold has a metric tensor that satisfies the laws of Riemannian geometry, meaning that distance and angles are well-defined. On the other hand, a differentiable manifold not Riemannian does not have this metric tensor, making it a more general and abstract mathematical structure.
Examples of differentiable manifolds that are not Riemannian include topological manifolds, which are manifolds that are locally Euclidean but do not have a metric tensor, and smooth manifolds with non-positive curvature, such as hyperbolic spaces.
Studying differentiable manifolds that are not Riemannian allows for a more abstract understanding of mathematical structures and can be useful in fields such as differential geometry, topology, and physics. These types of manifolds also have important implications in understanding spaces with non-Euclidean geometries.
Yes, a differentiable manifold not Riemannian can be embedded in a higher dimensional space, just like a Riemannian manifold. However, the embedding may not preserve the metric or other geometric properties of the manifold, making it a non-isometric embedding.