List of formulas in Riemannian geometry Puns

One way to think about Riemannian geometry is to think of it as a generalization of the Pythagorean theorem: instead of having ds^2 = dx^2 + dy^2, we can have ds^2 = g_{ij} dx^i dy^j. However, we are still dealing with degree 2 polynomials, just like in the Pythagorean theorem.

Is there a reason why no one talks about generalizing this to higher-order polynomials? For example, can we generalize the Pythagorean theorem to have something like ds^2 = dx^3 + dy^2 with higher-order powers included?

I realize that from a physical point of view, units would be a problem, but that can easily be taken care of by including constants that allow change of units.

My question can be taken in at least two different directions, so I'll address what I think:

• I know there is something called Finsler geometry. Instead of having two inputs for the metric g(v, w), the metric only has one input g(v). Basically, you have distances, but no specific notion of orthogonality. I haven't looked into Finsler geometry in any serious way, but am I correct to say this is one way to answer my question? I assume it allows the function g(v) to be any polynomial in the components of v.

• Can we have something where the metric has three or more inputs? Instead of g(v, w), we have g(v, w, u) = g_{ijk} v^i w^j u^k. Are there any references that study this in any way?

Honestly, what my real question is, why is it "natural" to generalize the Pythagorean theorem to Riemannian geometry, but not further? Once you go beyond, you seem to start losing features instead of gaining them (like orthogonality). Is there something special to powers-of-2 / rank-2-tensors / degree-2-polynomials in geometry?

I'm studying some Riemannian geometry for a program, and it is going fairly smoothly so far. The thing that has struck me is that there are many more definitions and formulas than any other branch of math which I've studied previously. Of course, many of the formulas can be rederived from the definitions if one has enough patience with manipulating tensors, but this can be a rather tedious and lengthy process to go through, especially if it is only an intermediate step as part of a much longer problem. For those of you that use a fair amount of geometry in your work, to what extent do you memorize formulas vs look up the relevant formula? Do you have any formulas that you'd recommend prioritizing committing to memory (e.g. Levi-Civita I'm guessing)? I'm also open to any general suggestions for studying the subject (I have a solid background in manifold theory and the typical US undergraduate math major courses).

In Riemannian geometry, the model space is Euclidean space.

In CR geometry, the model space is the Heisenberg group.

Are there more general geometries based on different Lie groups? I am aware of so-called nilgeometry, solvgeometry, etc. But as far as I know, there isn't as developed a theory as CR and Riemannian.

edit: I'm feeling that Riemannian geometry is not quite the correct geometry to choose here... My loose understanding is that the Heisenberg group is a model for boundaries in C^(n+1). Of course Euclidean space servers a similar role.

They have curved space. Curved. Space.

ok so i have a couple of basic questions about the riemannian gradient. i have a chart \Phi from a subset of R^m to a submanifold of R^n. (let's just say the subset of R^m is a tangent space of the submanifold) I want to use it to relate the gradient of a function f in the chart (\nabla f) to its gradient (grad f) on the manifold. I found the metric tensor G by doing d \Phi ^T d \Phi and the inverse G^-1. I can find the Riemannian gradient by multiplying G^-1 * \nabla f at a point p in the tangent space. However, when i do this i get an m-dimensional vector. what basis is this with respect to? is it the standard basis for R^m except rotated to the tangent space at Phi(p)? im confused because i read the columsn of d \Phi are the basis of the tangent space.

also by the chain rule, i have \nabla f = d \Phi ^T grad f. this should give me grad f if i multiply \nabla f by the generalized inverse of d \Phi^T, right? but it gives me something totally different ):

my second question is just how do you compare grad f at different points on the submanifold aka how does parallel transport work. if im in the same chart, can i compare grad f at two different points by comparing G^-1 * \nabla f?

I study math but also have a profound interest in Deleuze's philosophy, and I find his writings about Riemannian geometry, metrics, manifolds, etc. fascinating, but I don't have an intuitive enough understanding of these concepts to fully understand what Deleuze is saying when he references these ideas.

For reference, I know what a manifold is, I have some idea what tensors and metrics are, and I have a vague understanding of what Riemannian geometry and differential geometry studies.

I think I somewhat understand the formalism.

There's one valid way i know if how to think about the Parallel Transport. If the manifold (eg a sphere) is embedded in R^(n) then: To transport u from p along v to p', I simply move u while a force normal to sphere acts on it to keep it tangent to the sphere throughout. The resulting u' at p' is what we wanted.

But this is 3D, the picture is complicated. How do I draw Parallel Transport of the sphere in 2D? >!I want to do something similar to this: 1. Chose an orthogonal basis that includes v. Draw some geodesics creating a grid that will serve as coordinates. If u at p is an arrow pointing 2 squares to the left and 1 up of p in the grid. Then the transported u' at p' will just be the arrow with the same coordinates p'. I say similar because this doesn't work.!<

Hi all,

I am not a mathematician. I studied physics and would like to learn more about Riemannian geometry (I didn't get to hear the lectures back at Uni...) but I can't seem to find anything online which is not already too advanced for me. Especially, I am struggling to find examples, exercises etc.

Does anyone have any recommendations?

Thanks!

From what I understand, "Riemannian Geometry" is an integral part of Einstein's General Theory of Relativity. But what exactly is it? And why is the fabric of space time considered Riemannian by nature? Thanks!

If so could you give me some examples?

(Since reddit doesn't lay out nicely with plain text/LaTeX, I've tried to keep the TeX to a minimum.) Let U^(m) = R^(m-1) x R^(+) and B^(ρ) be the open ball of radius ρ in R^(m). Let Φ:U^(m) → B^(ρ) be a diffeomorphism given by

[; \Phi(x,y) := \rho \left(\frac{-2x}{\|x\|^2 + (y+1)^2}, \frac{\|x\|^2 + y^2 - 1}{\|x\|^2 + (y+1)^2} \right) \in \mathbb{R}^{m-1} \times \mathbb{R} = \mathbb{R} ;]

where [; (x,y) \in \mathbb{R}^{m-1} \times \mathbb{R}^{+} = \mathbb{U}^{m} ;].

I'm to show that geodesics in the Poincare ball are given by semicircles (or straight lines through the origin) that are orthogonal to the boundary.

I know that geodesics in U^(m) are parameterized by

[; \gamma(t) = (x(t),y(t)) = \left( R \cos(2 \arctan(e^t)) \xi + a , R \sin(2 \arctan^(e^t))\right) ;]

where R is the radius of the circle centered at a and ξ is a unit vector in R^(m-1). By isometry, this means that Φ(γ(t)) is a geodesic in B^(ρ). I already showed that the "x-axis" (for lack of a better description) maps to the boundary of B^(ρ), so orthogonality is preserved by isometry, but how do I show that Φ(γ(t)) is a semicircle?

Any help is greatly appreciated.

Thank you.

Hey, I'm stuck on the proof of a formula for the Gauss curvature of a regular surface on $R^3$ with an orthogonal parametrization.

The statement: (It's actually the exercise 4.3.1 in P. do. Carmo's Differential Geometry of Curves and Surfaces.)

>$S$ is a regular surface in $R^3$. $X: (u,v) \in U \mapsto X(u,v) \in S $ is an orthogonal parametrization of $S$ of a neighbour of a point $p \in S$. (i.e. $ X_{u} = \partial{X} / \partial{u}$ and $X_{v} = \partial{X} / \partial{v}$ are orthogonal.) Let $E, F, G$ be the coefficients of the first fundamental form of $S$. Then the Gauss curvature $K$ of $S$ at $p$ is
>
>$$K = -\frac{1}{2\sqrt{EG}}\left[ \left(\frac{E_v}{\sqrt{EG}}\right)_{v} + \left(\frac{G_u}{\sqrt{EG}}\right)_{u} \right].$$

I know that we can first compute the Christoffel symbols and use the equation $X_{uuv} = X_{uvu}$ to deduce this formula, but this method seems too complicated and doesn't explain the meaning of the terms $\frac{E_v}{\sqrt{EG}}$ and $\frac{G_u}{\sqrt{EG}}$.

I want to find a proof that is more direct and find out some geometric meanings of the formula (if possible), but I have no clue.

I have found that if we take $e_1 = X_{u}/|X_{u}|$ and $e_2 = X_{v}/|X_{v}|$, then the inner product of $(e_2)_u$ and $e_1$ is exact $\frac{E_v}{2\sqrt{EG}}$ and the inner product of $(e_1)_v$ and $e_2$ is exact $\frac{G_u}{2\sqrt{EG}}$. But I'm not sure whether this is relevant to the formula.

I want to prove this formula because it can be used to simplify the proof the local Gauss-Bonnet theorem.

So because I wanted to know what the non-VATS DPS (including the average crit chance and reload time) is of all weapons, I made part of this list a long while ago on a Google Document. I just got the idea to complete and share this list due to a comment I just received on another post: https://www.reddit.com/r/fo3/comments/ivzxg3/share_your_fallout_3_100_completion_stats_here_is/

The Fallout Wiki does give an extended DPS combat formula, but that list is not all too accurate and well-ordered in my opinion: it doesn´t take all perks, attacks per second and time reloading into account. Plus, it also only considers sneak-damage: https://fallout.fandom.com/wiki/Fallout_3_combat

So first I´ll share how I calculated mine: ((Base Damage + xeno, pyro, Superior Defender) x projectiles x attacks per second) + (Crit damage x Better Criticals x projectiles x critical chance (standard is 10% since Luck = 10) + 5% (Finesse) = 15% (so f.e. 2x therefore is 2 x 15% = 30%) x attacks per second) x (ammo cap. / (attacks/sec x ammo/shot)) / ((ammo cap. / (attacks/sec x ammo/shot)) + reload time).

The slightly more comprehensible formula is as follows: ((Base Damage + 3 perks x projectiles) x attacks per second) + (Crit damage x Better Criticals x projectiles x crit chance including Finesse x attacks per second) x (clip size / (attacks per second x ammo/shot)) / ((clip size / (attacks per second x ammo/shot)) + reload time).

So basically, I´m simply adding the total base damage to the total average crit damage, and then calculating those damages by a certain second period, including the reloading part (or in other words, the time spent shooting rather than reloading).

Luckily, the Fallout Wiki indeed already displayed most of these numbers per weapon page, but the end results were still very incomprehensible, including the long list of the link I gave above. The in-game pip-boy damage is also pretty misleading, because that formula is way less comprehensive.

From now on, the only important part is the perks I included in the non-VATS damage formula, since there are many ways to do so. I decided to calculate the end-game DPS, on a character with 10 Luck after taking the Almost Perfect perk plus the Bobblehead thereafter. I left the parts of the calculation in my end results so others are fre

Hi, I am a Musicology Masters doing research into the aesthetic motivations for Neo-Riemannian transformations, and I was wondering if anyone hear had some pop music examples that they could share. I would like to build a decent dataset and any help would be appreciated. Examples I already found were Blow up the World by Soundgarden, Dinosaur by King Crimson and Easy meat by Frank Zappa. Thanks in advance!

Hi there. I am sure this is something simple. I have a list of tasks assigned to my team with a column to specify which employee is assigned the task and a column to show what the current status is (not started, in progress, complete). I want to show how many tasks each employee has broken down by their current status. I have worked out the formula to count how many tasks are assigned to each person, and how many tasks are at each status but I can't seem to combine them. Thanks in advance!

I'm looking for a simple way to list every week in a year as text using the structure [Week of June 28 - July 5] . Beginning Monday and ending Sunday. One week part row in a single column. TIA!

I put Geometry Dash in #1 because it's my absolute most favorite game above any other. What about you guys? What's your #1 game?