Differential geometry Definition and 177 Discussions
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
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...
Hello everyone,
I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor.
I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇ρδgμν if δ{gμν} is...
Hello,
does anyone know an (more or less) easy differential geometry book for courses in generall relativity and quantum field theory? I'm looking for a book without proofs that focus on how to do calculations and also gives some geometrical intuition. I already looked at The Geometry of...
Hi, I'm trying to calculate the second fundamental form of a circle as the boundary (submanifold) of a spherical cap. I'm not sure if I'm doing it right. Is it possible to do that without parametrize the manifolds?
I wrote the parametrization of the spherical cap (which is the same as the...
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...
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 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...
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...
Homework Statement
Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
Homework Statement
Show that the metric connection transforms like a connection
Homework Equations
The metric connection is
Γ^{a}_{bc} = \frac{1}{2} g^{ad} ( ∂_{b} g_{dc} + ∂_{c} g_{db} - ∂_{d} g_{bc} )
And of course, in the context of Einstein's GR, we have a symmetric connection,
Γ^{a}_{bc}...
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...
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...