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.
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?
Reading the English translation of Einstein's seminal paper on GR.
http://einsteinpapers.press.princeton.edu/vol6-trans/90?ajax
This paragraph below on p78 doesn't make much sense to me.
Could you provide a second English translation or even adding math notation.
"Before Maxwell, the laws...
I would like to begin my first exploration of the arts of differential geometry/topology with the first volume of M. Spivak's five-volume set in the different geometry. Is a thorough understanding of vector calculus must before reading his book? I read neither of his Calculus nor Calculus on...
I don't know how to create a commutative diagram here so I'd like to refer to Diagram (1) in this Wikipedia article. I need to discuss the application of this diagram to the tangent bundle of a smooth manifold because there are some basic points that are either glossed over or conflict in the...
Hello,
do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)?
This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology).
Thanks in advance!
While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz...
Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric:
So we have Ricci flow equation,∂tgμν=-2Rμν.
And we have metric tensor for schwarzschild metric:
Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...
Hello,
I will be enrolling in an undergraduate Classical Mechanics course and I was wondering if the book by Spivak "Physics for Mathematicians: Mechanics" would help me understand the concepts more in depth than usual.
Until the time that I will be taking the course, I will already have...
Hi, there is a book of dg of surfaces that is also about tensor calculus ?
Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus!
Thank you in advance
On chapter over regular surfaces, In definition 1 point 2. He says that x: U → V∩S is a homeomorphisms, but U⊂ℝ^2 onto V∩S⊂ℝ^3. I am confused, how can it be so!
As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...
Homework Statement
This is a problem from Spacetime and Geometry by Carroll,
Just because a manifold is topologically nontrivial doesn't necessarily mean it can't be covered with a single chart. In contrast to the circle ##S^1##, show that the infinite cylinder ##RxS^1## can be covered with...
I'm fairly new to differential geometry (learning with a view to understanding general relativity at a deeper level) and hoping I can clear up some questions I have about coordinate charts on manifolds.
Is the reason why one can't construct global coordinate charts on manifolds in general...
Hi, I want to find the number of parameters needed to define an orthogonal transformation in Rn.
As I suppose, this equals the dimension of the orthogonal group O(n,R) - but, correct me if I'm wrong.
I haven't been able to figure out how to do this yet. If it helps, I know that an orthogonal...
I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the...
Hello everyone, I've 2 books on manifolds theory in e-form:
1) Spivack, calculus on manifold
2) Munkres, analysis on manifold
What would be good to begin with? :oldconfused:
Thank you in advance
Goodmorning everyone,
is there any implies to use in general relativity a metric whose coefficients are harmonic functions?
For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function?
In (1+1)-dimensions is well-know that the Einstein...
Let ##V## be a quaternionic vector space with quaternionic structure ##\{I,J,K\}##. One can define a Riemannian metric ##G## and hyperkahler structure ##\{\Omega^{I},\Omega^{J}, \Omega^{K}\}##. Do this inner product
$$\langle p,q \rangle :=...
I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions:
1) Wald first defines...
In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...
Hello!
I want to learn about the mathematics of General Relativity, about Topology and Differential Geometry in general. I am looking for a book that has applications in physics. But, most importantly, i want a book that offers geometrical intuition(graphs and illustrations are a huge plus) but...
Hi all, I need some help- I was reading Carroll's GR book, and on pages 71-71 he discusses the metric in curved spacetime. I have a few questions regarding this section:
(1) He says
In our discussion of path lengths in special relativity we (somewhat handwavingly) introduced the line element...
I'm looking for a demonstration of the equation governing the evolution of the scalar field:
## \Box \phi = \frac{1}{\sqrt{g}} \frac{ \partial}{\partial x^{\mu}} \sqrt(g)g^{(\mu)(\nu)} \frac{\partial}{\partial x^{\nu}} \phi=0##
I used the lagrangian for a scalar field: ## L = \nabla_{\mu}\phi...
I am trying to prove the following:
$$3d\sigma (X,Y,Z)=-\sigma ([X,Y],Z)$$
where ##X,Y,Z\in\mathscr{X}(M)## with M as a smooth manifold. I can start by stating what I know so it is easier to see what I do wrong for you guys.
I know that a general 2-form has the form...
Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically...
Homework Statement
As the title says, I need to show this. A conformal transformation is made by changing the metric:
##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}##
Homework Equations
The Weyl tensor is given in four dimensions as:
##...
I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative.
As I understand it a connection ##\nabla...
I need help to visualize the geometry involved here,
How can I visualize the last paragraph? Why is the surface of fixed r now an ellipsoid? Also for r = 0, it is already a disk? I've tried searching for the geometry of these but I can't find any image of the geometry that I can just stare...
As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
I am studying differential geometry and I stumbled on something that I don't understand. When we have a m- dim differential manifold, with U_i and U_j open subsets of M with their corresponding coordinate
function phi. As can be seen in the figure.
If I understand it correctly phi_j of a...
Consider the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##, a point belonging to the envelope of this map satisfy the condition ##J_{\phi}(t,s)=0##. What is the role of the Jacobian in maps like these and why points in the envelope have to satisfy ##J_{\phi}(t,s)=0##?
I am taking my bachelor in geometric quantization but I have no real experience in differential geometry ( a part of my project is to learn that). So I find myself in need of some good books that cover that the basics and a bit more in depth about symplectic manifolds.
If you have any...
Hi, I'm new here and I'm trying to learn GR. I wanted to know the math books that I need to tackle GR properly, so far the books that I came across are:
Tensor Analysis on Manifolds by Bishop and Goldberg
Tensors, Differential Forms, and Variational Principles by Lovelock and Rund
I have a good...
1. Homework Statement
I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...
How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...
I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames...
On this picture we see a octagonal dome. I am trying to calculate the volume of this object by integral calculus but I can't find a way. How would you calculate this?
https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202015-12-17%20kl.%2002.14.48.png [Broken]
I am majoring in...
I am having a hard time understanding vector transformations. I know that vectors must transform a certain way and that dual vectors (or covectors) transform the "opposite" way. What is strange to me is that the basis vectors transform like dual vectors and the basis dual vectors transform like...
Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...
I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...
I have an equation that says $$C_1\partial_{\mu}G^{\mu\nu}+C_2\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}G_{\rho\sigma}=0$$ If I want to get rid of the ##\epsilon^{\mu\nu\rho\sigma}## in the second term, I know I must multiply the equation by some other ##\epsilon## with different set...
Hi,
Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t
but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
I thought about the Taylor expansion on a Riemannian manifold and guess the Taylor expansion of ##f## around point ##x=x_0## on the Riemannian manifold ##(M,g)## should be something similar to:
f(x) = f(x_0) +(x^\mu - x_0^\mu) \partial_\mu f(x)|_{x=x_0} + \frac{1}{2} (x^\mu - x_0^\mu) (x^\nu -...
Homework Statement
I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks
Homework Equations
I am trying to derive the curvature...