differential geometry

Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...

Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.

I am trying to derive the radial momentum equation in the equatorial Kerr geometry obtained from the equation $$ (P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad $$. Expressing the first term in the equation as $$ (P+\rho)u^\nu u^r_{;\nu}=(P+\rho)u^r u^r_{;r} $$ I obtained the...

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...

My name is Martin Scholtz and I am a postdoc researcher at the Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. I'm working mainly in the area of gravitational physics, but I am interested in different topics as well, see tags...

Let us consider Ashtekar's definition of asymptotic flatness at null infinity: I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition. In fact, a distinct approach to asymptotic flatness already starts with...

For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds: $$ df = \sum^n \frac{\partial f}{\partial x_i} dx_i $$ If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...

Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...

Hi everyone! I have a problem with one thing. Let's consider the Lorentz group and the vicinity of the unit matrix. For each ##\hat{L}## from such vicinity one can prove that there exists only one matrix ##\hat{\epsilon}## such that ##\hat{L}=exp[\hat{\epsilon}]##. If we take ##\epsilon^{μν}##...

1. Homework Statement This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function 2. Homework Equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$ 3. The Attempt at a...

Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!

Hello, In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as $$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...

This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...

Hi, consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...

Hello I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you

Hi, Let ##M^3## be a 3-manifold embedded in ##\mathbb R^3## and consider a 2-plane field ( i.e. a Contact Structure) assigned at each tangent space ##T_p##. I am trying to understand obstructions to defining the plane field as a 1-form ( Whose kernel is the plane field/ Contact Structure) Given...

I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates. In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following: In the previous sections, flat...

O'Neill's Elementary Differential Geometry, problem 4.3.13 (Kindle edition), asks the student to show that the image of an open set, under a proper patch, is an open set. Here is my attempt at a solution. I do not know if it is complete as I have difficulty explaining the consequence of the...

O'Neill's Elementary Differential Geometry contains an argument for the following proposition: "Let C be a curve in a plane P and let A be a line that does not meet C. When this *profile curve* C is revolved around the axis A, it sweeps out a surface of revolution M." For simplicity, he...

O'Neill's Elementary Differential Geometry, in problem 3.4.5, asks the student to prove that isometries preserve covariant derivatives. Before solving the problem in general, I decided to work through the case where the isometry is a simple inversion: ##F(p)=-p##, using a couple of simple vector...

I ran across exercise 2.8.4 in Oneill's Elementary Differential Geometry. It says "Given a frame field ##E_1## and ##E_2## on ##R^2## there is an angle function ##\psi## such that ##E_1=\cos(\psi)U_1+\sin(\psi)U_2##, ##E_2=-\sin(\psi)U_1+\cos(\psi)U2## (where ##U_1##, ##U_2##, ##U_3## are the...

Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...

I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...

I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says: ."""" ........Of course, this in itself does not tell us why the Clifford bundle has no continuous cross-sections. To understand this it will be helpful to...

Let $$\phi(x^1,x^2....,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$ However the answer is given using metric tensor which I am not able to derive. Here is the...

1. Homework Statement This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the...

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold. My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a...

I am watching these lecture series by Fredric Schuller. [Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00 In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold. He shows...

I try to solve the following problem: If S be submanifold of M and every smooth function f on S has a smooth extentsion to all of M, then S is properly embedded. [smooth means C-infinity]. I can show that S is embedded. What I need is to show either S is closed in M or the inclusion map is...

Reading Chandrasekhar's The mathematical theory of black holes, I reached the point in which the Newman-Penrose GR formalism is explained. Actually I'm able to grasp and understand the usage of null tetrads in GR, but The null tetrads used in this formalism, are very special, since are made by...