Differential geometry Definition and 409 Threads

Problem1.3. Describe the one-sheeted hyperboloid as a surface of revolution; that is, find a positive function f : R \rightarrow R such that x(u, v)= \left[ \begin {array}{c} f \left( u \right) {\it cos}\nu \\\noalign{\medskip}f \left( u \right) {\it sin}\nu \\\noalign{\medskip}\nu\end...

Hi all, I'm taking an introduction to general relativity course along with an elementary differential geometry course this term. I'm really interested in this stuff and I've been waiting 3.5 years to take these courses, so I'm really excited. Which textbooks have you all come across that...

This is not homework: I was wondering if there was a website that gave the solutions to A Comprehensive Introduction to Differential Geometry by Michael Spivak. I was learning this on my own. NOT homework.

Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0. How can I show that a Euclidean circle centered at the origin is a hyperbolic circle?

http://www.maths.adelaide.edu.au/michael.murray/dg_hons/ Contents Co-ordinate independent calculus. Introduction Smooth functions Derivatives as linear operators. The chain rule. Diffeomorphisms and the inverse function theorem. Differentiable manifolds Co-ordinate charts...

Hi everyone. This is kind of a geometry/quantum mechanics question (hope this is the right place to post this). So, in quantum mechanics we consider operators that reside in an infinite dimensional Hilbert space (to speak rather informally). We even have the cool commutator relation, which is...

Hi, I've decided to learn GR myself recently since it's like the "sexy" side of physics. But I'm getting stuck with the tensors notations already. Maybe my math background is just not sufficient enough to do GR. In general, how do I know that an object is tensorial; for example, objects like...

I wonder if there are some relationships between the torsion in algebra and the torsion in differential geometry. Could someone tell me something about them?

what are some good places, schools, to study differential geometry in terms of physics?

Hello all, I am taking a class on differential geometry and I have run into a problem with the following question: Show that if α is a regular curve, i.e., ||α'(t)|| > 0 for all t ∈ I, then s(t) is an invertible function, i.e., it is one-to-one (Hint: compute s'(t) ). I am not really...

So I've taken two differential topology/geometry classes both from a mathematics department. I see all over this forum a whole lot of talk about indices being up or down and raising/lowering etc. My professors barely ever mentioned these things though I did notice that when they worked in...

I go to a small liberal arts university that only offers certain math classes at certain times. Due to the way my schedule has worked out, I only have the option of taking ONE of the following: geometry, DEs, or DG. What should I do? By the time I HAVE to chose, I will have taken the calculus...

Homework Statement Can some one please explain to me how to show that J^{\alpha}{ }_{;\alpha}={1\over{\sqrt{-g}}}\partial_\alpha(\sqrt{-g}J^\alpha)Homework Equations \Gamma^\gamma{}_{\alpha\beta}={1\over 2}g^{\gamma\delta}(g_{\delta\alpha,\beta}+g_{\delta\beta,\alpha}-g_{\alpha\beta,\delta})...

I would just like to know which of these math courses is best suited for physics. I have taken advanced calculus and linear algebra, so I've seen most of the proofs one typically sees in an intro analysis course (ie. epsilon delta etc.). I intend to do work with a lot of Quantum Field Theory...

Homework Statement Assume that \tau(s) \neq 0 and k'(x) \neq 0 for all s \in I. Show that a necessary and sufficient condition for \alpha(I) to lie on a sphere is that R^2 + (R')^2T^2 = const where R = 1/k, T = 1/\tau, and R' = \frac{dr}{ds}Homework Equations \alpha(s) is a curve in R3...

What are some books that fit this description? i.e. a simple introduction to the subject?

Hi I am a 2nd year Pure Mathematics undergraduate student. I have always coveted Differential Geometry as it seems extremely interesting but I do not know much about it. The prerequisites for this class are MAT 208 (Linear Algebra) and MAT 314 (Intermediate Analysis I). I am in Linear Algebra...

Please recommend some good books of differential geometry for a physics student. Thanks!

Hi, I have been reading some stuff about differential geometry (with the ultimate goal of trying to understand loop quantum gravity...) and something I have been having trouble with for awhile is the idea of a connection. Basically, I have yet to find a definition that actually enables me to...

differential geometry hypersurface problem - need help starting! [b]1. Homework Statement [/b Let f\in C^{\infty}(R\R^{2}) and let S be the set of points in R\R^{3} given the graph of f. Thus, s={{(x,y,z=f(x,y))|(x,y)\inR\R^{2}}. a) Show that this set of points can be viewed as a regular...

Hi all, I'm torn between taking complex analysis or differential geometry at the advanced third year level. Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics? I know that differential geometry shows up in general relativity...

Homework Statement On R^3 with the usual coordinates (x,y,z), consider the pairs of vector fields X,Y given below. For each pair, determine if there is a function f:R^3-->R with non-vanishing derivative df satisfying Xf=Yf=0, and either find such a function or prove that there is none. (a)...

Homework Statement For any open set U \subset \mathbb{R}^n and any continuous and injective mapping f : U \rightarrow \mathbb{R}^n, the image f(U) is open, and f(U) is a homeomorphism. Homework Equations N/A The Attempt at a Solution I am trying to learn how to write proofs, so...

I'm trying to teach myself differential geometry from the internet, and I've hit a snag in proving homeomorphisms. First, show that \Re^n is homeomorphic to any open ball in \Re^n. (I'm not sure how to write the conventional "R" using Latex.) I'm trying to prove this statement, but I am...

My understanding of GR is very coordinate oriented which kind of drags me down when I try to answer more general questions. Can somebody recommend a book in Differential Geometry with application to GR? Here are my preferences. I don't like hand waving typical for some books 'written for...

1) a. Show that if a curve C is a line of curvature and a geodesic then C is a plane curve. - Pf. Let a(s) be a parameterization of C by arc length (which I'm assuming always exists). then C is a geodesic if the covariant derivative of a(s) with respect to s is 0. We also know that if C is...

Ok, in general, I know that if the coefficients of the First Fundamental Form agree for two surfaces parameterized by X and Y, the the map X(Y^-1) is an isometry, or the two surfaces are isometric. I also know that if two parameterizations don't have the same coefficients, this does not imply...

I have to prove the map g:torus --> (S^2 3 dimensional sphere of radius 1) is a C infinity map in my assignment. The torus is parameterized as x(u,v)=(3+cos u)(cos v) y(u,v)=(3+cos u)(sin v) z(u,v)=sin u The map g is given by g=[6yz/(x^2+y^2+z^2+8), 3-sqrt(x^2+y^2), -xz/sqrt(x^2+y^2)] I...

What orginally motivated the field of differential geometry?

I want a book on introductory Diff. Geometry for self study, but I don't know which book will be suitable for me, and not on a too high level. I know Linear Algebra, single & multi variablecalculus, basic vector calculus and very little about ODE's and PDE's. At the Diff. Geometry course...

what is a good book in differential geometry. I currently know calculus, a bit about differential equations, a bit of linear algebra and a bit about tensors. I also know some variational calculus. Of course what I know won't really help. I've skimmed through some physics sources and...

Hi, I just want to know if I'm on the right track with these questions... 1. To prove that a point and a line in R^3 are not surfaces, I showed that the function from an open interval U in R^2 to the intersection of the point/line and a subset W in R^3 cannot be a homeomorphism. This is...

I'm looking for a good book on riemannian geometry, with a minimum of prerequistes and that takes a more intutive rather than formal approach. I know a bit of calculus of variations, multivariable calculus, vector calculus, and a bit of linear algebra.

Homework Statement Let alpha(t) be a regular curve. Suppose there is a point a in R^3 space s.t. alpha(t)-a is orthogonal to T(t) for all t. Prove that alpha(t) lies on a sphere.Homework Equations Definations: A regular curve in R^3 is a function alpha: (a,b)-->R^3 which is of class C^k for...

There was a little problem in my final exam that went "Show that a conformal equi-areal map is an isometry". I invoqued the caracterisation of "conformal" that the two metrics are proportional, say by a proportionality function L: E1=LE2, F1=LF2, G1=LG2. Then I invoked the caracterisation of...

Does anyone know where I can find a nice set of notes accompanied with pictures on Tensor Analysis or Differential Geometry?

Does anyone know where I can find a nice set of notes accompanied with pictures on Differential Geometry?

Should I take a semester-long differential geometry course (in lieu of a thermo/stat mech course, which while not terribly challenging would prepare me for qualifiers, etc), or should I just jump into GR and maybe pick up a mathematical methods in physics course later? I first learned partial...

One often encounters the following identitiy in Tensor Analysis/Differential Geometry: dx^j (partial/partial x^i) = partial x^j / partial x^i = delta ij It's easy to see why partial x^j / partial x^i = delta ij but how does dx^j (partial/partial x^i) = partial x^j/partial x^i ? I...

I've just started learning about intrinsic equations after learning about envelopes, arc length and curved surface area in cartesian, parametric and polar coordinates, and understand the principle behind intrinsic equations, but my book doesn't explain it particularly well, here is a question...

So, where do I begin? I have a current interests in going into Algebraic Topology and I am currently learning about Point-Set Topology and Abstract Algebra, which are pre-requisites to Algebraic Topology. After talking to a professor, he recommended learning some Differential Geometry, but...

I'm looking for resources of worked problems in modern differential geometry. There are plenty that have theorems and general formulas but very few that gon into the dirty details of how to solve actual problems. Any information would be appreciated.

I am a senior in HS right now. I am taking my classes at a local university because I finished my HS classes early, but will still graduate with my class this June. I am taking Differential Equations now, but would like to take a geometry class next semester. The prereqs are multivariable...

differential geometry... I am constantly being told that this is important stuff for undergradautes to know about and it is taught by the end of the secon year...however, I am not going to be seeing ANY of this stuff at all during undergraduate school. And there isn't even a class offered in...

I saw GRTensorII mentioned here awhile back. What are the most powerful tools out there (which still run on a PC), including user manuals and texts, that you would recommend for working with field equations, curvature tensors, line elements, etc?

I'm trying to demonstrate the following proposition: Let \vec{\alpha}(s) be a natural parametrization of an arc C. Then: \vec{\alpha}(s+h)=\vec{\alpha}(s)+\left(h-\frac{\kappa^2h^3}{6}\right)\hat{t}+\frac{1}{2}\left(\kappa...

I want to learn the mathematical language of Ashtekar's formulation / Loop Quantum Gravity in 3 months. Which introductory differential geometry texts do you recommend? I prefer books that are more mathematical, such as those that start from definitions, theorems, proofs, etc but not too...

Tensors & Differential Geometry -- What are lie groups? I've heard a lot about "lie groups" on this section of the forum, and was wondering what they are and if someone could explain it in simple terms. Thank you.

Hello, I was wondering what you guys think is the absolute minimum requirements for learning Differential Geometry properly and also how would you go about learning it once you got to that point, recommended books, websites, etc. I am learning on my own because of some short circuit in my brain...

What are the best books for learning differential geometry well? Any recommendations appreciated.