# What is Riemannian geometry: Definition and 30 Discussions

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

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1. ### I Diverging Gaussian curvature and (non) simply connected regions

Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...
2. ### Riemann curvature coefficients using Cartan structure equation

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...
3. ### I Riemannian Fisher-Rao metric and orthogonal parameter space

Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
4. ### B Dilating or expanding a closed ball in Riemannian geometry

Hello. If a closed ball is expanding in time would we say it's expanding or dilating in Riemannian geometry? better saying is I don't know which is which? and what is the function that explains the changes of coordinates of an arbitrary point on the sphere of the ball?
5. ### A Riemannian Geometry: GR & Importance Summary

Hi I've done a masters taught module in GR and from what I've learned these are two of some of the most important significance of needing a Riemannian Geometry: 1) If we consider the Lagrangian of a freely-falling particle given by ##L= \int ds \sqrt{g_{uv}\dot{dx^u}\dot{dx^v}} ## and find the...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### 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...
10. ### 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)...
11. ### A Proving the Differential Map (Pushforward) is Well-Defined

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...
12. ### 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...
13. ### 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...
14. ### 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.
15. ### 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...
16. ### 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...
17. ### Riemannian Metric Tensor & Christoffel Symbols: Learn on 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.
18. ### Deriving Riemann Tensor Comp. 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...
19. ### 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
20. ### Levi-Civita Connection & Riemannian Geometry for GR

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...
21. ### Riemannian Geometry exponential map and distance

Hi all, I was wondering what the relationship between the Riemannian Geometry exponential map and the regular manifold exponential map and for the reason behind the name.
22. ### Learn Riemannian Geometry: Resources for Self-Learners

Can someone recommend some background texts which can build me up with the necessary pre-requisites to learn about Riemannian Geometry? I have been self studying single and multi variable calculus but lack the mathematical rigour. Some resources/textbooks that can cover the background material...
23. ### Geometry Riemannian Geometry by Do Carmo

Author: Manfredo Do Carmo Title: Riemannian Geometry Amazon link https://www.amazon.com/dp/0817634908/?tag=pfamazon01-20 Prerequisities: Basic differential geometry, topology, calculus 3, linear algebra Level: Grad Table of Contents: Preface How to use this book Differentiable...
24. ### Horizontal Lift vs Parallel Transport in Principal Bundle & Riemannian Geometry

I am a physicist trying to understand the notion of holonomy in principal bundles. I am reading about the horizontal lift of a curve in the base manifold of a principal bundle (or just fiber bundle) to the total space and would like to relate it to the "classic" parallel transport one comes...
25. ### Riemannian Geometry is free of Torsion. Why use it for General Relativity?

As I understand it, Riemannian geometry doesn't allow Torsion (a property of geometry involving certain permutations among the indices of Christoffel Symbols). Does this restrict the geometry of General Relativity (GR) to describing only a curved spacetime with the Riemann curvature tensor? Is...
26. ### Riemannian Geometry: Is It a 2nd Year Grad Course?

Hey folks, Is it generally true that for US Math PhD programs, Riemannian geometry is a 2nd year grad course? I was looking at JHU, one of the PhD programs I will be applying to, and they don't require you take any prereq grad courses. And the cirriculum seems to be the standard RG...
27. ### How do you prove immersion? (Basic Riemannian Geometry)

Homework Statement the question i have is more of a conceptual question, i have no idea how to prove that a mapping would be an immersion. thus i have no clue how to start the assigned problem here's the specific problem: prove that (e~) is an immersion : note (e~ means phi tilda) Let F...
28. ### Coordinates in Riemannian Geometry

Hi, I was wondering if Geodesic polar coordinates, Geodesic shperical coordinates and Riemann Normal coordinates are the same. Also, are there any standard techniques for computing these coordinates for a manifold given in terms of level set of a function. Are there any good references that...
29. ### Proving Homeomorphism is a Diffeomorphism | Riemannian Geometry

Hello. Let M,N be a connected smooth riemannian manifolds. I define the metric as usuall, the infimum of lengths of curves between the two points. (the length is defined by the integral of the norm of the velocity vector of the curve). Suppose phi is a homeomorphism which is a metric...
30. ### Some basic problems in Riemannian Geometry

Hi all! I just found this site today, and I am really hoping that I can get some useful advice here. That said, I have two problems--one easy, one not so easy. Easy problem: Basically, I was wondering if anybody out there knows of an algorithm to calculate g_{ij} , given only the...