differential geometry

1. I Geodesics subject to a restriction

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 $\Delta t=t_{1}-t_{0}$...
2. I Curve inside a sphere

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.
3. A Derivation of radial momentum equation in Kerr geometry

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...
4. I Understanding the definition of derivative

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...
5. Hello!

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...
6. A BMS coordinates near future null infinity

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...
7. I Understanding vector differential

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...
8. Riemann Curvature Tensor in 2D

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...
9. A Killing vectors corresponding to the Lorentz transformations

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 $\epsilon^{μν}$...
10. Differential 1 form on line

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...
11. I Is the commutator of vector fields an important notion?

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!

26. Question about Spherical Metric and Approximations

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...
27. A Constructing a sequence in a manifold

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...
28. A How are curvature and field strength exactly the same?

I am watching these lecture series by Fredric Schuller. [Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller] @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...
29. A Properly embedded submanifold

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...
30. I Geometric meaning of complex null vector in Newman-Penrose

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...