Remarkable clarity by Dr. Frederic. Although I am thoroughly familiar with differential geometry still the 21 lectures so far have refreshed many fine points. Specially Lie groups and associated Lie algebras as an applied tool hinted during many lectures was one of the best aspects of this entire lecture series. Nice blend of abstraction with applications in physics.
One remark: The vertical subspace, defined as the kernel of the push forward of the bundle projection map, does not lie in the fibre (or is the fibre) as mentioned (12:10), but it is tangent to the fibre.
Now I get why principal bundles are considered a natural generalization of Riemannian manifolds! Actually, I'm not used to the applications in physics you've given here, but still I got much insight from you. Thanks a lot, sir!
This and the previous lecture, Schuller talks about the existence of a section a couple of times. However, I feel like we haven't treated this or I missed it. Is there a good place to read up on this? I believe Schuller introduced sections without the requirement of them being continuous or smooth. Without these requirements, it seems like they would always exist.
The lecture introduces the notion of a connection one-form w_p as a map: w_p: T_p P ---> T_e G. How can this be since a one-form is a (anti-symmetric) tensor and should hence yield a number in R ( rather than a vector in T_e G )? It seems that the output of w_p describes the 1-form itself rather than the output of a 1-form. In that sense w_p would be a map from a vector in T_p P into the cotangent space of T_e G.
There is something I also do not understand about this because I usually think of one forms as maps from the cotangent space at a point to the real numbers. But somehow this idea never shows up.
Well it s actually a Lie Algebra valued one form so the output is actually vector valued , i would recommend to look up the definition on a vector bundle and then generalize it so on a vec tor bundle an ( affine) connection is a map from the sections over a bundle E to to the sections of cotangent bundle tensored with the bundle itself . In my course we defined it using a frame and looking up how the connection with respect to a vector field acts on it yielding a vetor field with coefficients being the entries of the connection form . The definition of this course does not depend on that but it can be at first a bit bizarre to work with but it s actually really easy .
It's what we call in mathematics an abuse of notation. This is not your standardly defined 1-form, in a sense it simply what it is. But many of the properties of 1-forms also hold for these objects, hence by a stretch of notation the name Lie-algebra-valued 1-form.
04:00 I think I am confused about the order of the approach, from connection to parallel transport and covariant derivative. In Riemann Geometry, we define the connection. Then we use it to define covariant derivative and so we can define parallel transport. Am I misunderstand the approach in Riemann Geometry? If not, why we have the different approaches?
Many thanx for these amazing lectures. Please, let me pose a question. At 17:30 - 21:35, the professor gives an intuitional picture of the Horizontal subspace as a linear choice that completes the whole tangent space TP, together with the vertical subspace VP and hence have a Whitney Sum. Later, until 28:56, the formal definition is given. I was wondering where the assumption of linearity of the connection is hidden in these definitions. And more specifically, would it be meaningful to assume a non-linear connection which is compatible with these definitions?
On a fibre bundle which in general is not a vector bundle, a connection can be viewed as a Whitney complement to the vertical bundle. So you can take any bundle complementary to the vertical bundle and declare it as the horizontal bundle. There is no further linearity involved.
The 'right action of Horizontal subspace' seemed unnatural to me. Then I read on nLab that the condition of 'right action (push forward) of Horizontal space' in the connection can be motivated by producing a group homomorphism between the fibres. I did not understand this. How do I prove the homomorphism using Prof. Schuller's definition?
32:48 That picture is convincing that the horizational subspace can change Hor(X) but it looks like the vertical projection remained the same from the picture. It's inconsistent.
As an illustration, take usual 2d space and suppose I have the vector v = (1,1) in the basis (e1, e2) where e1 = (1,0) and e2 = (0,1), i.e. v = 1 * e1 + 1 * e2. Now, keep e1 as is, and instead take e2' = (1,1), then clearly v = 1 * e2' = (0,1) in that basis. As you can see from that example, it is actually the component associated with e1 (i.e. the vector that did not change) that actually changed! Mathematically, by linear independence, there is a unique decomposition of a vector into a given basis. If you change a single basis vector, the decomposition is different and there is no reason to assume that only the component associated with the new basis vector changes. From a picture perspective, the reason is that when you want to find the component of a vector wrt to some basis vector, you have to project parallel to said basis vector. In 32:48, to find the vertical component, you then have to project the vector parallel to (i.e. along) H_p P (and not parallel to the blackboard as you do) towards V_p P. This projects it to the point p and so the associated component is 0.
