What is Differential geometry: Definition and 418 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. Jianphys17

    Differential Geometry book with tensor calculus

    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
  2. B

    Courses Should I take a course in differential geometry?

    Hi guys, I'm thinking of maybe of studying differential geometry as part of my undergraduate degree. However, it's not for physicists, it's a full on formal mathematics course specifically for mathematicians. I'm not sure whether it's a bit overkill and won't actually be useful. We don't have a...
  3. Jianphys17

    I Do Carmo's book, chap2 Regular surfaces, definition 1.2 -- question

    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!
  4. F

    A A question about coordinate distance & geometrical distance

    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...
  5. F

    Infinite cylinder covered by a single chart

    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...
  6. F

    A Manifolds: local & global coordinate charts

    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...
  7. JuanC97

    I Dimension of the group O(n,R) - How to calc?

    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...
  8. F

    A What is a topology intuitively?

    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...
  9. Jianphys17

    Introduction book to Differential Geometry

    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
  10. A

    A Metric with Harmonic Coefficient and General Relativity

    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...
  11. L

    A Hyperhermitian condition

    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 :=...
  12. L

    A Hyperhermitian inner product -- explicit examples requested

    Can anyone show me explicit examples of Hyperhermitian inner product?
  13. J

    A Covariant derivative definition in Wald

    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...
  14. F

    I How to interpret the differential of a function

    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...
  15. J

    Geometry Book on Differential Geometry/Topology with applications

    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...
  16. G

    A Meaning of ds^2 according to Carroll

    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...
  17. V

    A Evolution of Scalar Field: Equation Demonstration

    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...
  18. F

    I Prove what the exterior derivative of a 3-form is....

    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...
  19. Markus Hanke

    I Geometric Interpretation of Einstein Tensor

    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...
  20. A

    Weyl Tensor invariant under conformal transformations

    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: ##...
  21. D

    A Confusion on notion of connection & covariant derivative

    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...
  22. Jianphys17

    Prerequisites for non Euclidean geometry

    Hi, i would be very interested to start learning hyperbolic geometry, what would be the necessary prerequisites to begin it's study? :smile:
  23. W

    Visualizing the space and structure described by a metric

    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...
  24. D

    Non-Euclidean geometry and the equivalence principle

    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...
  25. F

    Understanding the Purpose of Charts in Differential Geometry

    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...
  26. V

    Geometry Spivak's differential geometry vs calculus on manifolds

    Hi, I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first...
  27. M

    Rotations in differential geometry

    Simple and basic question(maybe not). How are rotations performed in differential geometry ? What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially. I am looking to calculate the angle between two geodesics. Can this...
  28. Icaro Lorran

    Envelope of a parametric family of functions

    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##?
  29. F

    Topology Learn Differential Geometry: Books for Bachelor in Geometric Quantization

    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...
  30. W

    Relativity Mathematics book before General Relativity

    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...
  31. N

    Spivak & Dimension of Manifold

    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...
  32. M

    MHB What are the practical applications of differential geometry?

    Hey! :o In what jobs is differential geometry applied and needed?
  33. T

    A Opposite "sides" of a surface - Differential Geometry.

    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...
  34. S

    Contravariant and covariant vectors

    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...
  35. T

    Volume of an octagonal dome by using calculus

    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 I am majoring in math-econ but...
  36. Schwarzschild90

    Differential geometry : Tangent vector & reparameterization

    Homework Statement Problem statement uploaded as image. Homework Equations Arc-length function The Attempt at a Solution Tangent vector: r=-sinh(t), cosh(t), 3 Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use...
  37. N

    Difficulty understanding vector transformation law

    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...
  38. D

    Tangent spaces at different points on a manifold

    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...
  39. S

    Differential Geometry page 193 Nakahara

    Hey, I am struggling to understand what the following is in terms of the mathematics (see Nakahara page 193 at the bottom...
  40. D

    Dupin indicatrix differential geometry

    Hello 1. Homework Statement We define the Dupin indicatrix to be the conic in TPM defined by the equation IIP(v)=1 If P is a hyperbolic point show: a. That he Dupin indicatrix is a hyperbola b/ That the asymptotes of the Dupin indicatrix are given by IIP(v)=1 , i.e., the set of asymptotic...
  41. S

    General Relativity & Differential Geometry Q&A

    Dear all I am studying general relativity and i have a question as follow. We have the 2- sphere can be scanned totally by a coordinate system (theta, phi) with the metric tensor written in terms of theta and phi. Now i want to divide the 2-sphere into charts 4 charts then each will have its own...
  42. D

    Find the parameters of a curve (differential geometry)

    Hi, 1. Homework Statement C : ℝ→ℝ3 given by C(t)= ( 1/2 [ (1+k)/(1-k) cos((1-k)t) - (1-k)/(1+k) cos((1+k)t) ] ; 1/2 [ (1+k)/(1-k) sin((1-k)t) - (1-k)/(1+k) sin((1+k)t) ] ) with 0<|k|<1 Show that C(t) is an epitrocoid and find R, r and d according to k Homework Equations Parametrization of...
  43. D

    Ricci rotation coefficients and non-coordinate bases

    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...
  44. P

    Troubled with the indices

    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...
  45. S

    Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

    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.
  46. D

    Lie derivative of tensor field with respect to Lie bracket

    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...
  47. S

    General Relativity and Differential Geometry textbook problem

    I'm studying General Relativity and Differential Geometry. In my textbook, the author has written ##x^2=d(x,.)## where d(x,y) is distance between two points ##x,y\in M##. I couldn't understand what d(x,.) means. Moreover, I am not sure if this is generally true to write ##x^2=g_{\mu\nu} x^\mu...
  48. Kostik

    Which text on differential geometry to supplement relativity

    I am looking to pick up one of these texts, but I don't really want to buy all three. Is there a considered favorite? Thanks in advance. B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity T. Frankel: The Geometry of Physics B. Schutz: Geometrical Methods of Mathematical Physics
  49. S

    Riemannin generalization of the Taylor expansion

    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 -...
  50. P

    Checking derivation of the curvature tensor

    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...
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