Differential geometry Definition and 409 Threads

I've been reading Straumann's book "General Relativity & Relativistic Astrophysics". In it, he claims that the twice contracted Bianchi identity: $$\nabla_{\mu}G^{\mu\nu}=0$$ (where ##G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R##) is a consequence of the diffeomorphism (diff) invariance of the...

Let $h$ be a bump function that is $0$ outside $B_\epsilon^m(0)$ and posetive on its interior. Let $f$ be smooth function on $B_{2\epsilon}^m(0)$. Define $f^*(x)=h(x)f(x)$ if $x\in B_{2\epsilon}^m(0)$ and $=0$ if $x\in \mathbb{R^m}-B_\epsilon^m(0)$. I want to show that $f^*$ is smooth on...

I have been stuck several days with the following problem. Suppose M and N are smooth manifolds, U an open subset of M , and F: U → N is smooth. Show that there exists a neighborhood V of any p in U, V contained in U, such that F can be extended to a smooth mapping F*: M → N with...

So, I've been studying some tensor calculus for general theory of relativity, and I was reading d'Inverno's book, so out of all exercises in this area(which I all solved), this 6.30. exercise is causing quite some problems, so far. Moreover, I couldn't find anything relevant on the internet that...

The question: Show that the Lorentz condition ∂µAµ =0 is expressed as d∗ A =0. Where A is the four-potential and * is the Hodge star, d is the exterior differentiation. In four-dimensional space, we know that the Hodge star of one-forms are the followings. 3. My attempt Since the four...

I see that this has been discussed before, but the old threads are closed. As Carl Brans and others note, it seems too big a coincidence to ignore. Why is exotic smoothness "good" (in the sense that it permits richer physics or something like that)? Exotic Smoothness and Physics,arXiv "there...

Hello. In the following(p.2): https://michaelberryphysics.files.wordpress.com/2013/07/berry187.pdf Berry uses parallel transport on a sphere to showcase the (an)holonomy angle of a vector when it is parallel transported over a closed loop on the sphere. A clearer illustration of this can be...

This is my college's description for it: Differential Geometry (3) Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: Calculus IV, and Introduction to Linear Algebra; or consent. I was doing pretty well in all my...

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...

Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as Ad(U)ta = Λ(U)abtb I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...

1) Firstly, in the context of a dot product with Einstein notation : $$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$ with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...

I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...

Homework Statement Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2. Homework Equations General coordinate transformation, ds2=gabdxadxb The Attempt at a Solution I started with a general...

I am encountering this kind of problem in physics. The problem is like this: Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between...

Hello! I started reading some differential geometry applied in physics (wedge product, Hodge duality etc.) and how you can rewrite classical theories (Hamiltonian Mechanics, Electromagnetism) in a much nicer way. Can someone point me towards some reading about how can more information be...

Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...

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)...

Hi everyone! I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a...

I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...

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...

Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by: $$x = \sqrt U cos(V)$$ $$y = \sqrt U sen(V)$$ $$z = U$$ with the inverse relationship: $$V = \arctan \frac{y}{x}$$ $$U = z$$ The natural basis is: $$e_U = \frac{\partial \overrightarrow{r}}...

My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular. The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...

Hello! Can someone point me toward some (introductory) problems in differential geometry with solutions (preferably free)? Thank you!

I know two kinds formulas to calculate extrinsic curvature. But I found they do not match. One is from "Calculus: An Intuitive and Physical Approach"##K=\frac{d\phi}{ds}## where ##Δ\phi## is the change in direction and ##Δs## is the change in length. For parametric form curve ##(x(t),y(t))##...

Hi everyone I am reading Sean Carrol's lecture notes on general relativity. link to lecture : https://arxiv.org/abs/gr-qc/9712019 In his lecture he introduced dxμ as the coordinate basis of 1 form and ∂μ as the basis of vectors. I understand why ∂μ could be the basis of the vectors but not for...

Hello! I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)? For example, it would be great to have a technical textbook that explains the geometrical...

Good Day Early on, in Frankel's text "The Geometry of Physics" (in the introductory note on differential forms, in fact, on page 3) he writes: "We prefer the last expression with the components to the right of the basis vectors." Well, I do sort of like this notation and after reading a bit...

I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus. I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential...

I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...

In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...

Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...

Have members of the community had the experience of being taught GR both from a mathematical and physics perspective? I am a trained mathematician ( whatever that means - I still struggle with integral equations :) ) but I have always been drawn to applied mathematical physics subjects and much...

Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate...

Hello everyone, here I come with a question about inertial frames as defined in General Relativity, and how to prove that the general definition is consistent with the particular case of Special Relativity. So to contextualize, I have found that one can define inertial frames in General...

I am looking at the definition of the characteristic numbers from the wikipedia https://en.wikipedia.org/wiki/Characteristic_class#Characteristic_numbers "one can pair a product of characteristic classes of total degree n with the fundamental class" I am not sure how is this paring defined here...

Homework Statement [/B] This is a problem from my Differential Geometry course A velociraptor is spotting you and goes after you. There is a shelter in the direction perpendicular to the line between you and the raptor when he spots you. So you run in the direction of the shelter at a...

Hello, I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses...

Homework Statement The problem is described in the picture I've attached. It is problem number 6. Homework Equations Tangent line of a curve Length of a curve The Attempt at a Solution I don't know why I'm so confused on what seems like it should be a relatively straightforward problem, but I...

Homework Statement The problem statement is in the attached picture file and this thread will focus on question 7 Homework Equations The length of a curve formula given in the problem statement Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the...

I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions: Is such an abstract theory practical? I would say that homology is...

I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour. As I...

Consider an algebraic variety, X which is a smooth algebraic manifold specified as the zero set of a known polynomial. I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of X, or more precisely, of the Jacobian variety...

Hello, I have 2 questions regarding similar issues : 1*) Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle between the tangent vector and transported vector is...

I am looking at a statement that, for a short exact sequence of Abelian groups ##0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0## if ##C## is a free abelian group then this short exact sequence is split I cannot figured out why, can anybody help?

Hello! I try to think about the Foucault pendulum with the concept of parallel transport(if we think of Earth as being a perfect sphere) but I can't quite figure out what the vector that gets parallel transported represents(for example, is it the normal to the plane of oscillation vector?). In...

The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...

Hello I am interested in the Frenet-Serret Formulas (theory of curves?) relationship to theory of surfaces. 1) Can one arrive to the Frenet-Serret Formulas starting from the theory of surfaces? Any advice on where to begin? 2) For a surface that contain a space curve: if the unit tangent...

I am a physics student trying to self-learn Chern numbers and Chern class. The book I am learning (Nakahara) introduces the total Chern class as an invariant polynomial of local curvature form ##F## ## P(F) = \det (I + t\frac{{iF}}{{2\pi }}) = \sum\limits_{r = 0}^k {{t^r}{P_r}(F)} ## and each...

The material I am studying express the Ricci form as ##R = i{R_{\mu \bar \nu }}d{z^\mu } \wedge d{{\bar z}^\nu } = i\partial \partial \log G## where ##G## is the determinant of metric tensor, but I am not sure what does ##\log G## here, can anybody help?

Dear Physics Forum personnel, Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential...