riemannian geometry

1. A Is the Berry connection compatible with the metric?

Hello, Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)? Also, does it have torsion? It must either have torsion or not be...
2. A Is the Berry connection a Levi-Civita connection?

Hello! I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric). When performing a parallel transport with the L-C connection, angles and lengths are...
3. I Lie derivative of a metric determinant

I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric $g_{\mu\nu}$. The determinant of this metric is given by $g:=\text{det}(g_{\mu\nu})$. Given this, now consider the...
4. A On the dependence of the curvature tensor on the metric

Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...
5. Relativity Is Gravitation by Misner, Throne, Wheeler outdated?

Hi! With the re-release of the textbook "Gravitation" by Misner, Thorne and Wheeler, I was wondering if it is worth buying and if it's outdated. Upon checking the older version at the library, I found that the explanations and visualization techniques in the sections on differential(Riemannian)...
6. A Pushforward map

I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined. The map $$f:M\rightarrow N$$ is a smooth map, where $M, N$ are two smooth manifolds. If...
7. I How does parallel transportation relates to Rieman Manifold?

Source: Basically the video talk about how moving from A to A'(which is basically A) in an anticlockwise manner will give a vector that is different from when the vector is originally in A in curved space. $$[(v_C-v_D)-(v_B-v_A)]$$ will equal zero in flat space...
8. A Geometrical interpretation of Ricci and Riemann tensors?

I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...
9. A On embeddings of compact manifolds

I have found the following entry on his blog by Terence Tao about embeddings of compact manifolds into Euclidean space (Whitney, Nash). It contains the theorems and (sketches of) proofs. Since it is rather short some of you might be interested in.
10. Trouble understanding $g^{jk}\Gamma^{i}{}_{jk}$

Hi friends, I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of $g^{jk}\Gamma^{i}{}_{jk}$. I know it is a contracting relation on the Christoffel symbols and one can show that $g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})$ using the...
11. Ricci rotation coefficients and non-coordinate bases

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...
12. Metric Tensor in R2

Hi, Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
13. Deriving component form of Riemann tensor in general frame

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
14. Ricci tensor equals zero implies flat splace?

Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help