How to relate a one form to a vector.

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I don't quite get what you mean by there exists a natural inner product even when you don't have a metric because, if you have an inner product < , >, then |v|:= <v,v> (v : a vector) defines a natural metric. Of course, the converse is not true.

But when you define differential forms on a differentiable manifold, you by definition use inner product induced by trivialization. After that what kind of Riemannian (or Lorenzian or whatever) metric you endow on the manifold you have is up to you. And this additional structure is sometimes called PHYSICS, isn't it? (At least, I think. :P )

In Newtonian mechanics, your differentiable manifold has the simplest metric.
In Special Relativity your differential manifold has Lorenzian metric and called the Minkowski space.
In Quantum mechanics, it is still unknown, right?
As for the case involving temperature or other physical quantities, like momentum (in case of Hamiltonian mechanics), I don't know. Aren't they studying this kind of things in Dynamical System?

When you use diff mfds in physics to yield new results, since it is basically an application of a mathematical theory, I think the manifold corresponding to your physical phenamena should have a nice mathematical theory, or it will be just a new form of writing down equations. (I am not specializing on differential geometry, so I really cannot say such a thing, but I think if you don't fix an inner product of a manifold, the technique you can use to investigate it is very limited.)
 
errata:
Plz replace the first paragraph with this:

x,y are vectors. Define a metric d( , ) by d(x,y):=<x-y,x-y>.
 
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I don't quite get what you mean by there exists a natural inner product even when you don't have a metric because, if you have an inner product < , >, then |v|:= <v,v> (v : a vector) defines a natural metric.
I think I see the trouble you have with my explanation above. By inner product, you seem to be thinking of the product between two vectors. This does require a metric (or a bilinear form) in order to have an inner product.

There does however exist a natural inner product between vectors [tex]v \in T(M)[/tex] and covectors [tex]w \in T^*(M)[/tex] which are sometimes referred to a dual spaces. If [tex]dx_1 , dx_2 , ... , dx_n[/tex] is a covector basis ([tex]T^*(M)[/tex]), then the dual vector basis is [tex]\frac{\partial }{\partial x_1} , \frac{\partial }{\partial x_2} , ... , \frac{\partial }{\partial x_n}[/tex] with the natural inner product [tex]\langle dx_i , \frac{\partial }{\partial x_j} \rangle = \delta_{ij}[/tex] where [tex]\delta_{ij}[/tex] is the Kronnecker delta, or the identity matrix if you prefer. This inner product is inherent to the definition of the manifold, and does not depend on the addition of a metric to the manifold. I believe that is why Boothby refers to this as 'natural'.

But when you define differential forms on a differentiable manifold, you by definition use inner product induced by trivialization.
Previously you talked about an inner product between vector fields which requires a bilinear form or metric to be defined. Here you seem to be talking about the the natural inner product between vectors and covectors. Since these are two different types of inner products, it might be useful to separate the concepts and consider them independently since they are used for different purposes.

After that what kind of Riemannian (or Lorenzian or whatever) metric you endow on the manifold you have is up to you. And this additional structure is sometimes called PHYSICS, isn't it? (At least, I think. :P )
....
As for the case involving temperature or other physical quantities, like momentum (in case of Hamiltonian mechanics), I don't know. Aren't they studying this kind of things in Dynamical System?

When you use diff mfds in physics to yield new results, since it is basically an application of a mathematical theory, I think the manifold corresponding to your physical phenamena should have a nice mathematical theory, or it will be just a new form of writing down equations. (I am not specializing on differential geometry, so I really cannot say such a thing, but I think if you don't fix an inner product of a manifold, the technique you can use to investigate it is very limited.)
A manifold provides the basic structure for doing calculus in a much more general space than euclidean space. It provides coordinate systems, and exhibits topological properties of the underlying space. Some physical phenomenon depend on the topology of a space, so understanding them deos not depend on a metric, and in fact a metric may confuse the issue. For example chaos is studied effectively with topological approaches.

Of course a metric tends to be crucial to understand other laws. A metric is important to represent certain symmetries that are not captured by simply having a space with coordinate charts defined. For example, Newton's Law F=ma relies on a metric to convey that there is a symmetry between the x,y, and z directions. Capturing this symmetry is very important. However sometimes a nonmetric approach reveals more insight into a particular phenomena.
 
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I see. Thanks.

It is not essential, but I think calling a bilinear map (or perfect pairing) an inner product is not a standard use of term. As Boothby's notation is not standard.... Boothby is a nice textbook, though.

You mean in physics you often need to deal with topological, non differentiable, manifolds? It must be pretty tough because you can't use differential technique to study topological properties...or you probably meant topological invariants of manifolds play an important role in doing physics.

anyways, thanks.
 
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You mean in physics you often need to deal with topological, non differentiable, manifolds? It must be pretty tough because you can't use differential technique to study topological properties
I can't think of any physics problems I have dealt with that were topological in nature but non differentiable. I have dealt with various problems where topological invariants provided deep insights into the behavior of a system. I was originally attracted to some of the modern tools of differential geometry because they have proved to be so fruitful identifying topological invariants of physical systems (many of these properties are unrelated to the metric structure on a space). There seems to be a huge breadth and depth of topics that can be treated using these tools. I see so many new (to me) and interesting things done with these tools that I can only say I have scratched the surface, but the tools have proved to be very powerful in the problems I deal with.

or you probably meant topological invariants of manifolds play an important role in doing physics.
I have typically dealt with differential equations. I typically have been concerned with topological invariants of the differential equation, which are influenced by the topology of the underlying manifold. However, many of the topological invariants of interest are related to the specific differential equation you are dealing with.
 

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