- #1

bronxman

- 32

- 0

**Is the following correct?**

We begin with a set.

Then, we specify a certain collection of subsets and thereby create a topology. This endows the set with certain properties, one of which is “nearness” and “boundedness.”

Then we specify that the topology be smooth. In so doing, our topology become a manifold.

Finally, we specify a metric. The metric enables measurements: angles, distances, etc.

We now have a set on which we can perform calculus. And by perform “calculus,” I mean study how things change, ON THE SET. And, more specifically: tangent vectors, tangent bundles, etc.

Now, this is different from how we performed calculus in R3. In R3, we have all the rules of calculus. But a manifold restricts R3, except in small local regions where it is still like it. (Not to mention that the coordinates of a manifold could be temperature, pressure, etc.: all different.. whatever that means.)

They could be, say, rotation matrices...

Now we move onto Lie Groups. A Lie Group is a manifold because it is a continuous group. So once we map, say, the nine components of a matrix, and realize that if the matrix is orthogonal, then we have 3 independent coordinates. We can use the Euler angles.

And now, in the tangent space to the manifold we have a linear space. And now we can use Lie Algebra.

AND THAT is where I have the last problem.

__What does it mean to replace the Lie Group with the Lie Algebra?__Finally, as an aside, in the PROCESS of learning manifolds, I now understand exterior algebra. And from that, forms. And from forms, Stokes theorem. But while that is beautiful in itself (seeing the nature of the divergence, gradient and curl, along with the Green’s theorem and calculus of variations as a special case of the generalized Stokes), I do NOT need exterior forms if I am to understand dynamics from the perspective of manifolds and lie derivatives.

(This does not mean I understand... rather it means I finally know what I do not know...and where I can now begin learning.)

And if you can add to this statement in ANY WAY.. . PLEASE...