I am self studying differential geometry (and information geometry), and came across the following idea from a lecture by Frederic Schuller:
https://youtu.be/2eVWUdcI2ho?t=384
Essentially, one can define an abstract sphere via, $(S^2, \mathcal{O}, \mathcal{A})$, where $\mathcal{O}$ is the topological strucutre, and $\mathcal{A}$ the atlas, of abstract sphere structure $S^2$. From here it is claimed that through one's choice of linear connection (covariant derivative), one can define:
- The standard round sphere with one choice of covariant derivative, $\nabla$, so we have structure: $(S^2, \mathcal{O}, \mathcal{A}, \nabla)$
- Or an ellipsoidal structure with a slightly different chosen connection $(S^2, \mathcal{O}, \mathcal{A}, \nabla')$
Thus it would appear that one's choice of connection solidifies the geometric shape of the topological manifold (is this intuition correct so far?)
My main confusion arises when studying information geometry. In this frame work one can define an $\alpha$-connection as follows:
Methods of Information Geometry pg 32
And in this framework it can be shown that the $\nabla^{(1)}$ ($\alpha=1$), and $\nabla^{(-1)}$ ($\alpha=-1$) connections result in $\nabla^{(1)}$-flat and $\nabla^{(-1)}$-flat geometries. An example statement is taken from the text here:
Page 35
Thus with specific choice of $\alpha=1$ and $\alpha=-1$ one arrives at a flat connection, and notions such as parallel transport become meaningful. However for the self-dual case of $\alpha=0$ which defines the Levi-Civita connection over information geometries there are left over Cristoffel symbols thereby rendering a non-affine connection over the information manifold ** (upon reading this statement, this appears to be incorrect but I will leave it in, in case it helps people pinpoint the sources of my confusion). **
I am trying to resolve this latter idea, with the former, since connections appear to be tied strongly together with ones choice of covariant derivative (connection). Thus:
Main Question:
In the first video the simple change of the connection / covariant derivative has changed the underlying geometric representation of the smooth manifold (sphere to ellipsoid). Thus in information geometries if we continuously vary $\alpha$ from $\alpha=1$ through to $\alpha=0$ then to $\alpha=-1$, is it true that the underling smooth manifold is also changing in shape? Because the represented covariant derivative is changing also? Or is the shape static and flat (since we know in the $\alpha=1$ and $\alpha=-1$ scenarios one has a $\nabla$-flat manifold, so it seems weird for a geometric structure to transform from flat, to non-flat, to flat).
On some level I think this not to be the case, and that the information geometry more so is simply changing co-ordinate representation and it remains the same shape throughout and is globally flat. However, then I cannot resolve the misunderstanding in the first linked video, wherein a change in connection does seem to changing the underlying geometric shape.