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I’m coming from an engineering background so I’ve very little capacity to understand the abstract things. I decided to self study differential geometry to understand the General Relativity. With the help of online community and various visually illustrated books, I reached at the Tangent Bundle two years ago. Then I tried to learn the concept of connection, parallel transport, covariant derivative. But my brain can’t comprehend these things without proper visual aid & explanation. Somebody please help me out to break the two years deadlock.

What I know so far:(might be wrong though)

  1. In tangent bundle(TM), the tangent space(TpM) at each point acts as a Fiber.TM has a projection map π:TM ↦M here inverse π is the fiber. Section is the map s:M ↦TM which somehow indicates a vector field on M for tangent bundle TM.
  2. Now we are interested to connect two different fibers or compare two different vectors on M. But we need additional structure because two different points on M have different vector spaces. We consider a curve γ on M and we’d like to lift this curve from M to TM in some good manners(like vectors respecting parallelism or something). For that we need a connection, mainly Ehreshmann connection or levi Civita connection.
  3. What this connection does is: deconstructing the tangent space of total space TM into vertical and horizontal space.
  4. We can think of the vertical space as consisting of those vectors tangent to the fibers(i can imagine this for line fiber but can’t imagine for 2D manifold) and it projects to the base point in the manifold.
  5. The connection tells us what the horizontal space look like or how it behave. From this point I am lost.

My few questions:

  1. For circle as M, the tangents can be considered as fibers. It is easy to visualize TM. Just flip the tangent lines as vertical lines,Which might look like a cylinder. Now, tangent space(lets call it TTM) for the whole space TM at a point (p,v) is: I think 2 dimensional as TM is 2 dimensional. Now if we want to decompose the TTM by introducing something named connection like this: TTM= HTM+VTM. We think VTM as the vectors(lives in TTM) tangent to the fiber which points directly to base point p in M. Now the connection tell us how the HTM is oriented or something. I am having a hard time understanding how the connection fixes how the HTM is oriented and how this orientation gives us parallel transport or covariant derivatives. Please explain this.
  2. I can’t just comprehend how the TTM & its decomposition(HTM &VTM) will look like for 2D base manifold. As base M is 2D, TM will be 4D and TTM will be 4D too, i guess. In this scenario, HTM is 2D? and VTM is 2D? If it’s true then here VTM is also consists vectors pointing to the base point p at M. But what does this VTM is tanget to the fibers means?
Manzurul Hasan
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    If $\dim M=2$, then $\dim (TM)=4$ and $\dim (TTM)=8$, and for any splitting, $\dim(HTM)=\dim(VTM)=4$, so no you won’t be able to fully visualize this (for our everyday understanding of the term ‘visualize’). So, really, I’d suggest you just give up on that hope. But, if you’re willing to accept pseudo-visualization where you suppress dimensions in the drawings, then I refer you to this picture at the bottom of my question, and also this answer of mine for multiple descriptions of connections. – peek-a-boo Aug 01 '24 at 05:45
  • and maybe you might find this answer helpful after the other two (but it’s not quite as directly related to your current question). – peek-a-boo Aug 01 '24 at 05:48
  • sorry, typo in the first comment: we have $\text{rank}(HTM)=\text{rank}(HTM)=2$ as vector bundles over the 4-dimensional base manifold $TM$, so $\dim(HTM)=\dim(VTM)=4+2=6$. – peek-a-boo Aug 01 '24 at 06:01
  • See as well here for connections. – Jean Marie Aug 01 '24 at 06:25

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