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. D

    I Differential Geometry: Comparing Metric Tensors

    Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?
  2. M

    Covariant derivative of a (co)vector field

    My attempt so far: $$\begin{align*} (\nabla_X Y)^i &= (\nabla_{X^l \partial_l}(Y^k\partial_k))^i=(X^l \nabla_{\partial_l}(Y^k\partial_k))^i\\ &\overset{2)}{=} (X^l (Y^k\nabla_{\partial_l}(\partial_k) + (\partial_l Y^k)\partial_k))^i = (X^lY^k\Gamma^n_{lk}\partial_n + X^lY^k{}_{,l}\partial_k)^i\\...
  3. M

    The sphere in general relativity

    I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have $$\begin{align*}(\psi^+_1)^{-1}:\begin{array}{rcl} \{\lambda^1,\lambda^2\in [a,b]\mid (\lambda^1)^2+(\lambda^2)^2<1\}&\longrightarrow& \{\pm x_1>0\}\subset \mathbb{S}^2\\...
  4. M

    I Geodesics subject to a restriction

    Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...
  5. S

    Geometry Differential Geometry: Book on its applications?

    Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably...
  6. Celso

    I Curve Inside a Sphere: Differentiating Alpha

    Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.
  7. abby11

    A Derive Radial Momentum Eq. in Kerr Geometry

    I am trying to derive the radial momentum equation in the equatorial Kerr geometry obtained from the equation $$ (P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad $$. Expressing the first term in the equation as $$ (P+\rho)u^\nu u^r_{;\nu}=(P+\rho)u^r u^r_{;r} $$ I obtained the...
  8. L

    I Understanding the definition of derivative

    As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
  9. Martin Scholtz

    What Research Does Martin Scholtz Conduct in Gravitational Physics?

    My name is Martin Scholtz and I am a postdoc researcher at the Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. I'm working mainly in the area of gravitational physics, but I am interested in different topics as well, see tags...
  10. B

    A I need some fun questions with answers in differential geometry ()

    I am throwing a bachelor party for my brother, who is currently getting his PhD in Math at columbia, and as you might expect, he is not very much of a party animal. I want to throw him a party he’ll enjoy, so I came up with scavenger hunt in the woods, where every step in the scavenger hunt is a...
  11. L

    A Construct BMS Coordinates near Null Infinity

    Let us consider Ashtekar's definition of asymptotic flatness at null infinity: I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition. In fact, a distinct approach to asymptotic flatness already starts with...
  12. L

    I Understanding vector differential

    For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds: $$ df = \sum^n \frac{\partial f}{\partial x_i} dx_i $$ If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...
  13. W

    Riemann Curvature Tensor in 2D

    Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...
  14. wafelosek

    A Killing vectors corresponding to the Lorentz transformations

    Hi everyone! I have a problem with one thing. Let's consider the Lorentz group and the vicinity of the unit matrix. For each ##\hat{L}## from such vicinity one can prove that there exists only one matrix ##\hat{\epsilon}## such that ##\hat{L}=exp[\hat{\epsilon}]##. If we take ##\epsilon^{μν}##...
  15. Abhishek11235

    Is Every Differential 1-Form on a Line the Differential of Some Function?

    Homework Statement This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function Homework Equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$ The Attempt at a Solution...
  16. Zhang Bei

    I The Commutator of Vector Fields: Explained & Examples

    Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!
  17. ZuperPosition

    Abstract definition of electromagnetic fields on manifolds

    Hello, In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as $$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...
  18. V

    Geometry Classical and modern differential geometry

    Im planning on taking a course on classical differential geometry next term. This is the outline: The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium. The textbook is "differential...
  19. Gene Naden

    I How to prove that compact regions in surfaces are closed?

    This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...
  20. K

    Geometry Vargas' book about Differential Geometry

    I'm learning Differential Geometry (DG) on my own (I need it for robotics). I realized that there are many approaches to DG and one is Cartan's, which is presented in Vargas's book. I think that book is highly opinionated, but I don't know if that's a good or bad thing. Does anyone of you know...
  21. cianfa72

    I Differential structure on a half-cone

    Hi, consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...
  22. aboutammam

    I About the properties of the Divergence of a vector field

    Hello I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you
  23. W

    A Defining a Contact Structure Globally -- Obstructions?

    Hi, Let ##M^3## be a 3-manifold embedded in ##\mathbb R^3## and consider a 2-plane field ( i.e. a Contact Structure) assigned at each tangent space ##T_p##. I am trying to understand obstructions to defining the plane field as a 1-form ( Whose kernel is the plane field/ Contact Structure) Given...
  24. L

    A Constructing Bondi Coordinates on General Spacetimes

    I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates. In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following: In the previous sections, flat...
  25. Gene Naden

    I Showing that the image of an arbitrary patch is an open set

    O'Neill's Elementary Differential Geometry, problem 4.3.13 (Kindle edition), asks the student to show that the image of an open set, under a proper patch, is an open set. Here is my attempt at a solution. I do not know if it is complete as I have difficulty explaining the consequence of the...
  26. Gene Naden

    I Differential for surface of revolution

    O'Neill's Elementary Differential Geometry contains an argument for the following proposition: "Let C be a curve in a plane P and let A be a line that does not meet C. When this *profile curve* C is revolved around the axis A, it sweeps out a surface of revolution M." For simplicity, he...
  27. Gene Naden

    I Do Isometries Preserve Covariant Derivatives?

    O'Neill's Elementary Differential Geometry, in problem 3.4.5, asks the student to prove that isometries preserve covariant derivatives. Before solving the problem in general, I decided to work through the case where the isometry is a simple inversion: ##F(p)=-p##, using a couple of simple vector...
  28. Gene Naden

    I Connection forms and dual 1-forms for cylindrical coordinate

    I ran across exercise 2.8.4 in Oneill's Elementary Differential Geometry. It says "Given a frame field ##E_1## and ##E_2## on ##R^2## there is an angle function ##\psi## such that ##E_1=\cos(\psi)U_1+\sin(\psi)U_2##, ##E_2=-\sin(\psi)U_1+\cos(\psi)U2## (where ##U_1##, ##U_2##, ##U_3## are the...
  29. Avatrin

    Motivating definitions from differential geometry

    Hi I have always had an issue with understanding the definitions used in mathematics. I need examples before I can start using and reasoning with them. However, with tensor products, I have been completely stuck. Stillwell's Elements of Algebra was that made abstract algebra "click" for me...
  30. Bill2500

    I Topology vs Differential Geometry

    Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...
  31. shahbaznihal

    I Solving Tensor Calculus Question from Schutz Intro to GR

    I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
  32. Abhishek11235

    A Penrose paragraph on Bundle Cross-section?

    I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says: .""""...Of course, this in itself does not tell us why the Clifford bundle has no continuous cross-sections. To understand this it will be helpful to look at...
  33. S

    A Number Line in Synthetic differential geometry

    Hello! I just start looking at SDG and I'm already having difficulties with a few concepts as expressed by A Kock as: "We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R" "The geometric line can, as soon as one chooses two...
  34. Abhishek11235

    A Finding the unit Normal to a surface using the metric tensor.

    Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
  35. W

    Question about Spherical Metric and Approximations

    Homework Statement This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the...
  36. J

    A Constructing a sequence in a manifold

    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...
  37. V

    A How are curvature and field strength exactly the same?

    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...
  38. J

    A Properly embedded submanifold

    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...
  39. R

    I Geometric Meaning of Complex Null Vector in Newman-Penrose Formalism

    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...
  40. JuanC97

    I Issues with the variation of Christoffel symbols

    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...
  41. U

    Geometry Differential Geometry Book

    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...
  42. A

    A How to calculate the second fundamental form of a submanifold?

    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...
  43. D

    I Diffeomorphism invariance and contracted Bianchi identity

    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...
  44. J

    A Smoothness of multivariable function

    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...
  45. J

    A Can Smooth Functions be Extended on Manifolds?

    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...
  46. Antarres

    Null curve coordinate system

    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...
  47. QuantumRose

    About the derivation of Lorentz gauge condition

    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...
  48. Giulio Prisco

    A Physical meaning of "exotic smoothness" in (and only in) 4D

    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...
  49. J

    A Berry phase and parallel transport

    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...
  50. Eclair_de_XII

    Courses What kind of class is differential geometry?

    This is my college's description for it: Differential Geometry (3) Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: Calculus IV, and Introduction to Linear Algebra; or consent. I was doing pretty well in all my...
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