What is Differential geometry: Definition and 419 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.

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  1. C

    Please verify my differential geometry results

    Homework Statement Q1) One way to define a system of coordinants for a Sphere S^2 given by x^2 + y^2 + (z-1)^2 = 1 is socalled stereographical projection \pi \thilde \{N} \rightarrow R^2 which carries a point p=(x,y,z) of the sphere minus the Northpole (0,0,2) onto the intersection...
  2. H

    Calculating the Area of a Circle on S^2 in the Spherical Metric

    I need help with this problem: given a cirlce on S^2 of radius p in the spherical metric, show that its area is 2pi(1-cos p)
  3. D

    Differential Geometry Question

    Homework Statement Find an explicit unit-speed non-degenerate space curve \vec{r}:(0,\infinity)\rightarrow\Re^{3} whose curvature and torsion \kappa,\tau:(0,\infinity)\rightarrow\Re are given by the functions \kappa(s)=\tau(s)=\frac{1}{s}. Homework Equations the only thing that I can think of...
  4. S

    Introductory differential geometry text

    What is a very good differential geometry introductory text? My only background is Calculus (spivak). however, I'm very interested in mastering differential geometry (at both the pure math and physics application level). Any recommendations?
  5. D

    Can Differential Geometry Solve This Challenging Curve Containment Problem?

    Homework Statement Let \sigma:I\rightarrow R^{3} be a non-degenerate unit speed curve, and R be a real number >0. Fix a value s_{0}\in I. Prove that: (There exists a center \vec{p}\in R^{3} such that \sigma(I)\subset S_{R}(p))\iff (There exists an angle \phi\in R such that, for all s\in...
  6. L

    How Can Vector Calculus Help Understand Differential Geometry in Terms of Forms?

    if \alpha, \alpha' \in \Omega^1. Rewrite the identity, d(\alpha \wedge \alpha')=d \alpha \wedge \alpha' - \alpha \wedge d \alpha' in terms of vector calculus. I have absolutely no idea what is going on here. So if anybody could explain to me a) what this is all about and b) how to go about...
  7. L

    Differential Geometry Question

    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...
  8. M

    Exceptional books on GR and Differential Geometry

    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...
  9. F

    Find Solutions for A Comprehensive Introduction to Differential Geometry

    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.
  10. D

    Differential Geom: Showing Hyperbolic Circle from Euclidean Circle

    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?
  11. malawi_glenn

    Differential Geometry. Honours 1996

    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...
  12. Q

    Differential geometry in quantum mechanics - conserved quantities

    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...
  13. Q

    Tensors and differential geometry

    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...
  14. N

    Exploring Torsion: Algebra vs. Differential Geometry

    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?
  15. I

    Differential geometry + physics, departmental question

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

    Differential Geometry: Show Regular Curve is Invertible

    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...
  17. Cincinnatus

    Indices in differential geometry

    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...
  18. B

    Geometry, Differential Equations, or Differential Geometry

    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...
  19. P

    Differential geometry question

    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})...
  20. H

    Real Analysis vs Differential Geometry vs Topology

    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...
  21. M

    Differential Geometry Question

    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...
  22. P

    Differential geometry for dummies?

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

    Differential Geometry without Analysis?

    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...
  24. P

    Differential Geometry for General Relativity

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

    Connections in differential geometry

    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...
  26. Q

    Differential geometry hypersurface problem - starting

    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...
  27. J

    Easier to self-teach: differential geometry or complex analysis

    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...
  28. D

    Non-Vanishing Derivative Functions for Vector Fields X and Y in R^3

    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)...
  29. S

    An attempted proof of a theorem in elementary differential geometry

    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...
  30. S

    Elementary Differential Geometry Questions

    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...
  31. S

    A good book in Differential Geometry FOR GR?

    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...
  32. I

    Differential Geometry and geodesics

    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...
  33. I

    Differential Geometry general question

    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...
  34. H

    Local extension and differential geometry

    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...
  35. T

    Original motivation of differential geometry

    What orginally motivated the field of differential geometry?
  36. P

    Intro books on 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...
  37. T

    Differential geometry recommendations

    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...
  38. I

    Differential Geometry Homework

    (1) The gradient of a differential function f: S --> R is a differentiable map grad f: S --> R^3 which assigns to each point p in S a vector grad f(p) in the tangent space of p s.t. <grad f(p), v> at p = dfp(v) for every v in the tangent space Tp(S) (a) If E, F, G are the coefficients of...
  39. A

    Quick differential geometry questions

    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...
  40. P

    Involute and Evolute Proof (Differential Geometry )

    Homework Statement Suppose that a(s) is a unit speed curve. A) If B(s) is an involute of a (not necessarily unit speed), prove that B(s)= a(s)+ (c-s)*T(s), where c is a constant and T= a' B) Under what conditions is a(s) + (c-s)T(s) a regular curve and hence an involute of "a."...
  41. P

    Differential Geometry: Showing a curve is a sphere curve

    Homework Statement Show that a(x) =( -cos(2x)), -2*cos(x), sin (2x)) is a sphere curve by showing that (-1,0,0) belongs to each normal plane. Homework Equations Not quite sure (part of my question) T= a'(x) N= T'/norm(T') B= T x N (T cross N) The Attempt at a Solution Ok I found...
  42. R

    Good differential geometry book

    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.
  43. P

    Differential Geometry Proof (Need a Hand)

    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...
  44. quasar987

    Problem in my differential geometry final

    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...
  45. N

    Where to find notes onTensor Analysis or Differential Geometry.

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

    Discover the Beauty of Differential Geometry: A Visual Guide with Notes

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

    Should i take 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...
  48. M

    Proving an Identity from Differential Geometry

    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...
  49. R

    Defining Euclidean Norm in Phase Space: A Differential Geometry Analysis

    Hi to everynoe! I have a bit of trouble in understanding the following thing : Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time. Now, does it make a sense to define the euclidean norm of a vector in such...
  50. F

    Problem I'm having with intrinsic equations when studing differential geometry

    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...
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