Differential geometry Definition and 426 Threads

I would like to calculate the principal and Gaussian curvature of the spatial part of the Friedmann-Robertson-Walker (FRW) metric; specifically, the negative Gaussian curvature ##k=-1##. The FRW metric is, \begin{equation*} ds^2 = -dt^2 + R(t)^2 \left( \frac{dr^2}{1-k r^2} + r^2 d\Omega^2...

TL;DR Summary: How can I variate the quadratic Riemann curvature tensor, I tried raising and lowering the indices. Hi, Can you help me with this variation, I tried raising and lowering the indices. I tried for months every possible method to reach the following answer without success.

TL;DR Summary: In search of a suitable topic for an interesting undergraduate dissertation. I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential fields and...

I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential forms, chains, and a little bit of Theory of manifolds (Calculus on Manifolds, Michael Spivak). I am...

In a homework problem, I had to derive the relationship ##R_{\mu\nu} = \pm K g_{\mu\nu}## on a surface, i.e. a ##2##-dimensional submanifold of ##\mathbb{R}^3##. Here, ##K## is the Gaussian curvature. I think I managed to do that, but from my derivation I don't see why this result is restricted...

Following Steinacker's book we can say that given the manifold ##N## of a configuration space and its tangent bundle ##TN## we define a differentiable function ##L(\gamma,\dot\gamma): TN\rightarrow \mathbb{R}## and call it the Lagrangian function. We know there's always an isomorphism between a...

I would love to hear from you if you have any suggestions, feedback, or criticism. The goal is to build better and more sophisticated software that would push the boundaries of research in astrophysics!

In differential geometry, we typically define the boundary ##\partial M## of a manifold ##M## as all ##p \in M## for which there exists a chart ##(U,\varphi), p \in U## such that ##\varphi(p) \in \partial\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n = 0 \}##. Consequently, we also demand that...

Hi, From Lee book "Introduction on Smooth Manifolds" chapter 2, every topological manifold (Hausdorff, locally Euclidean, second countable) of dimension less then or equal 3 has unique smooth structure up to diffeomorphism. A smooth structure on a manifold is defined by a maximal atlas. So...

To my understanding, an orientation can be expressed by choosing a no-where vanishing top form, say ##\eta := f(x^1,...,x^n) dx^1 \wedge ... \wedge dx^n## with ##f \neq 0## everywhere on some manifold ##M##, which is ##\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n \geq 0 \}## here specifically. To...

Hello everyone, as part of my bachelor studies, I need to attend a seminar with the aim to prepare a presentation of about an hour on a certain topic. I have chosen the presentation about the generalized Stokes theorem, i.e. $$ \int_M d\omega = \int_{\partial M} \omega. $$ After hours of...

Hello everyone, we have recently covered electrodynamics in differential forms. I managed to get familiar with most of the concepts, but one thing came just up where I can't figure out what's going wrong. I tried computing the 3-form ##dx^i \wedge dx^j \wedge dx^k## by hand. However, even after...

I'm studying Liang's book on differential geometry for general relativity. There's a para in it talking about a maximal atlas - "Later on, when we talk about a manifold, we always assume that the largest possible atlas has been chosen as the differentiable structure, so that we can perform any...

I am working on this I am having trouble with b and c: b) Suppose ##(f_n)_{n=1}^{\infty}## is a sequence in ##Aut(G)##, such that ##(T_e(f_n))_{n=1}^{\infty} \to \psi## converges in ##Aut(\mathfrak g)## I want to show that ## f := \lim_{n\to \infty} f_n## exists as an continuous...

Does anybody know to which "Gauss embedding theorem" the speaker in this video talk at minute 14 (point 5. in the displayed notes) is refering too? Sounds to be a standard result in differential geometry, but after detailed googling I found nothing to which the speaker may refering too in the...

I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where...

TL;DR Summary: Reference request Hello! Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/books/differential-geometry-of-singular-spaces-and-reduction-of-symmetry/7D73498C35A5975594605428DA8F9267 I found that...

Hi, I'm keep studying The Road to Reality book from R. Penrose. In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...

In 3 d spherical coordinates we know that $$\triangledown \cdot \frac{\hat{\textbf{r}}}{r^2}=4π\delta^3(\textbf{r})$$ Integration over all## R^3## is 4π So when we remove the third dimensions and enter 2d polar coordinates then $$\triangledown \cdot...

Hi, I'd like to ask for clarification about the definition of differential of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##. As far as I know, the differential of a scalar function ##f## can be understood as: a linear map ##df()##...

Hi, suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it). Can its integral curves cross at any point ? Thanks. Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...

Any given sphere surface consists of a finite number of fixed points. If all these points on the surface were to rotate/flip in their locations by 180° in respect to the centre of the sphere simultaneously and hence making the entire sphere turn outside in, how do you go about formulating this...

##df=\frac {\partial f}{\partial r} dr+\frac {\partial f}{\partial \theta}d\theta\quad \nabla f=\frac{\partial f}{\partial r}\vec{e_r} +\frac{1}{r}\frac{\partial f}{\partial \theta }\vec{e_\theta }## On the other hand ## g_{rr}=1\:g_{r\theta}=0\:g_{\theta r}=0\;g_{\theta\theta}=r^2\;##So...

In fluid mechanics, it is sometimes useful to present the velocity, ##U## in terms of a scalar potential ##\Phi## as:##\vec{U}=\nabla \phi## ##U## has 3 dof. ##\phi## has 1.If asked why this works, in terms of a dof argument, why is this?e.g . compared to GR common decomposition of the Riemann...

I was reading Bernhardt Riemann's old foundational text on abelian functions, and I found a part that really confused me. What he is trying to do is set up an invariant to classify 2d surfaces as simply connected, multiply connected, etc via some kind of "connectivity number". From the text, I...

I am guessing that: $R_{a[bcd]}=0$ can not be derived from the symmetries of $R_{ab(cd)]}=R_{(ab)cd}=0$ $R_{[ab][cd]}=0$ ?Sorry when I search the proof for it I can not find much, it tends to come up with the covariant Bianchi instead. I am guessing it will need one of the symmetries above...

I am intending to join an undergrad course in physics(actually it is an integrated masters course equivalent to bs+ms) in 1-1.5 months. The thing is, in order to take a dive into more advanced stuff during my course, I am currently studying some of the stuff that will be taught in the first...

Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b},\\ \nabla_{a}n^{a}=0,\\ \nabla_{a}l_{b}=-\kappa n_{a}l_{b},\\ \nabla_{a}l^{a}=\kappa$$. How to show that under the variation of the null...

I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...

It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...

It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of...

Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold? Thank you!

Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math

School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...

This is from a series of lectures - "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller

I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...

What do we know of a curve(/what can it look like) in R^n if we know that κ1,κ2,...,κn-1 is constant?

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...

Hello everyone, I wanted some help deciding which elective to choose. I am a junior and for my next semester I have the option to pick either Differential Geometry-I or Quantum Information. I am confused which one to choose. We will be doing QMII as a compulsory course next semester and I have...

Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...

I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of the functions represent flux lines. I am trying to solve this problem in ## R^3 ## with a...

I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books. Among the others these two books by the same author seem to be the most widely recommended: Elementary Differential Geometry (Barret O' Neill) Semi-Riemannian Geometry with...

I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of...

Can anyone recommend a good on-line class for differential geometry? I'd like to start studying GR but want a good background in differential geometry before doing so. Many thanks.

In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function can't be equal to a constant and its partial differential be...

In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?

I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...

Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...

i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element 𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙 where 0≤𝜙<2𝜋 and 0≤𝜃<𝜋/2 Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...

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}...