Let M be a manifold and g a metric over M

[math6]

let M be a manifold and g a metric over M .
is it true that every subbundle from M must have the same metric g ?

 

[HallsofIvy]

Do you mean "sub bundle" or "sub manifold"? A (vector or otherwise) bundle over a manifold M is not a subset of M and cannot have the same metric.

A submanifold, on the other hand, is a subset of M with the topology inherited from M. If the topology on M is induced by a metric, then the topology on any sub manifold is induced by the same metric. The metric is inherited from M along with the topology.

 

[arkajad]

You can have a manifold with several different metrics on it. So, what exactly is your question? Subbundle of what? Of the frame bundle? Subbundle of the tangent bundle? Please describe, even if not quite precisely, what do you have in mind? What kind of an idea?

 

[mathwonk]

Maybe he means riemannian metric, i.e. a dot product on the tangent bundle, hence also on subbundles.

 

[math6]

sorry friends but i just now i see your answers .. thnx very much .
i mean submanifold .. but sorry HallsofIvy can you explain for me what do you mean exactly when you say "If the topology on M is induced by a metric " ??

 

[arkajad]

sorry friends but i just now i see your answers .. thnx very much .
i mean submanifold ..

This brings to my mind this question to which I do not remember seeing the answer: if we take a matrix group, say SU(2), is the invariant group metric the same as that induced by embedding in the the Euclidean space of 2x2 complex matrices? What about other groups like SU(1,1)?