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. F

    I Lie derivative of a metric determinant

    I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...
  2. Luck0

    A Characterizing the adjoint representation

    Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as Ad(U)ta = Λ(U)abtb I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...
  3. F

    I Demo of cosine direction with curvilinear coordinates

    1) Firstly, in the context of a dot product with Einstein notation : $$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$ with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
  4. F

    I Deduce Geodesics equation from Euler equations

    I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...
  5. jgarrel

    General relativity- Coordinate/metric transformations

    Homework Statement Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2. Homework Equations General coordinate transformation, ds2=gabdxadxb The Attempt at a Solution I started with a general...
  6. L

    A Can I change topology of the physical system smoothly?

    I am encountering this kind of problem in physics. The problem is like this: Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between...
  7. S

    I Differential geometry in physics

    Hello! I started reading some differential geometry applied in physics (wedge product, Hodge duality etc.) and how you can rewrite classical theories (Hamiltonian Mechanics, Electromagnetism) in a much nicer way. Can someone point me towards some reading about how can more information be...
  8. J

    A Is tangent bundle TM the product manifold of M and T_pM?

    Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...
  9. J

    Relativity Is Gravitation by Misner, Throne, Wheeler outdated?

    Hi! With the re-release of the textbook "Gravitation" by Misner, Thorne and Wheeler, I was wondering if it is worth buying and if it's outdated. Upon checking the older version at the library, I found that the explanations and visualization techniques in the sections on differential(Riemannian)...
  10. peroAlex

    Parametrize the Curve of Intersection

    Hi everyone! I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a...
  11. L

    A Can I find a smooth vector field on the patches of a torus?

    I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...
  12. F

    A Proving the Differential Map (Pushforward) is Well-Defined

    I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined. The map $$f:M\rightarrow N$$ is a smooth map, where ##M, N## are two smooth manifolds. If...
  13. Adrian555

    Natural basis and dual basis of a circular paraboloid

    Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by: $$x = \sqrt U cos(V)$$ $$y = \sqrt U sen(V)$$ $$z = U$$ with the inverse relationship: $$V = \arctan \frac{y}{x}$$ $$U = z$$ The natural basis is: $$e_U = \frac{\partial \overrightarrow{r}}...
  14. K

    Applied Differential geometry for Machine Learning

    My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular. The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...
  15. S

    I Problems in Differential geometry

    Hello! Can someone point me toward some (introductory) problems in differential geometry with solutions (preferably free)? Thank you!
  16. C

    I Extrinsic Curvature Formulas in General Relativity: Are They Equivalent?

    I know two kinds formulas to calculate extrinsic curvature. But I found they do not match. One is from "Calculus: An Intuitive and Physical Approach"##K=\frac{d\phi}{ds}## where ##Δ\phi## is the change in direction and ##Δs## is the change in length. For parametric form curve ##(x(t),y(t))##...
  17. Ron19932017

    I Why Denote 1 Form as dx? - Sean Carroll's Lecture Notes on GR

    Hi everyone I am reading Sean Carrol's lecture notes on general relativity. link to lecture : https://arxiv.org/abs/gr-qc/9712019 In his lecture he introduced dxμ as the coordinate basis of 1 form and ∂μ as the basis of vectors. I understand why ∂μ could be the basis of the vectors but not for...
  18. J

    Geometry Differential Geometry book that emphasizes on visualization

    Hello! I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)? For example, it would be great to have a technical textbook that explains the geometrical...
  19. JTC

    A Why Does Frankel Prefer Components on the Right in The Geometry of Physics?

    Good Day Early on, in Frankel's text "The Geometry of Physics" (in the introductory note on differential forms, in fact, on page 3) he writes: "We prefer the last expression with the components to the right of the basis vectors." Well, I do sort of like this notation and after reading a bit...
  20. Wrichik Basu

    Geometry Book Recommendations in Differential Geometry

    I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus. I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential...
  21. R

    [Symplectic geometry] Show that a submanifold is Lagrangian

    Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
  22. MattRob

    Showing that Metric Connections transform as a Connection

    Homework Statement Show that the metric connection transforms like a connection Homework Equations The metric connection is Γ^{a}_{bc} = \frac{1}{2} g^{ad} ( ∂_{b} g_{dc} + ∂_{c} g_{db} - ∂_{d} g_{bc} ) And of course, in the context of Einstein's GR, we have a symmetric connection, Γ^{a}_{bc}...
  23. L

    A Stokes' Theorem and Curvature on a Torus

    I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...
  24. L

    I Reference Frame Usage in General Relativity

    In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...
  25. joneall

    I Gradient one-form: normal or tangent

    Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...
  26. M

    I GR for a mathematician and a physicist? What's the difference?

    Have members of the community had the experience of being taught GR both from a mathematical and physics perspective? I am a trained mathematician ( whatever that means - I still struggle with integral equations :) ) but I have always been drawn to applied mathematical physics subjects and much...
  27. JuanC97

    I Understanding Co-vectors to Dual Spaces and Linear Functionals

    Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate...
  28. C

    A Inertial Frames: GR to SR | General Relativity

    Hello everyone, here I come with a question about inertial frames as defined in General Relativity, and how to prove that the general definition is consistent with the particular case of Special Relativity. So to contextualize, I have found that one can define inertial frames in General...
  29. L

    A Is there a natural paring between homology and cohomology?

    I am looking at the definition of the characteristic numbers from the wikipedia https://en.wikipedia.org/wiki/Characteristic_class#Characteristic_numbers "one can pair a product of characteristic classes of total degree n with the fundamental class" I am not sure how is this paring defined here...
  30. M

    A Velociraptor is pursuing you....

    Homework Statement [/B] This is a problem from my Differential Geometry course A velociraptor is spotting you and goes after you. There is a shelter in the direction perpendicular to the line between you and the raptor when he spots you. So you run in the direction of the shelter at a...
  31. B

    Courses Intro to Differential Geometry or in-depth PDE Course?

    Hello, I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses...
  32. M

    Verify Unit length to y-axis from Tractrix Curve

    Homework Statement The problem is described in the picture I've attached. It is problem number 6. Homework Equations Tangent line of a curve Length of a curve The Attempt at a Solution I don't know why I'm so confused on what seems like it should be a relatively straightforward problem, but I...
  33. M

    Proving length of Polygon = length of smooth curve

    Homework Statement The problem statement is in the attached picture file and this thread will focus on question 7 Homework Equations The length of a curve formula given in the problem statement Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the...
  34. L

    A Very basic question about cohomology.

    I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions: Is such an abstract theory practical? I would say that homology is...
  35. F

    A Diffeomorphisms & the Lie derivative

    I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour. As I...
  36. G

    A Period matrix of the Jacobian variety of a curve

    Consider an algebraic variety, X which is a smooth algebraic manifold specified as the zero set of a known polynomial. I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of X, or more precisely, of the Jacobian variety...
  37. F

    I Conservation of dot product with parallel transport

    Hello, I have 2 questions regarding similar issues : 1*) Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle between the tangent vector and transported vector is...
  38. L

    A A question about split short exact sequence

    I am looking at a statement that, for a short exact sequence of Abelian groups ##0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0## if ##C## is a free abelian group then this short exact sequence is split I cannot figured out why, can anybody help?
  39. J

    I Connection between Foucault pendulum and parallel transport

    Hello! I try to think about the Foucault pendulum with the concept of parallel transport(if we think of Earth as being a perfect sphere) but I can't quite figure out what the vector that gets parallel transported represents(for example, is it the normal to the plane of oscillation vector?). In...
  40. V

    I Infinitesimal cube and the stress tensor

    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...
  41. ltkach2015

    A Theory of Surfaces and Theory of Curves Relationship

    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...
  42. L

    A Why the Chern numbers (integral of Chern class) are integers?

    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...
  43. L

    A Ricci Form Notation: Need Help Understanding

    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?
  44. B

    Studying How to Learn both Differential Geometry and Relativity?

    Dear Physics Forum personnel, Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential...
  45. V

    A Geometry and integral laws of physics

    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...
  46. B

    Geometry Regarding to Spivak's Differential Geometry trilogy

    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...
  47. orion

    A Tangent Bundle questions about commutative diagram

    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...
  48. J

    Applied Books like J. Callahan's Advanced Calculus: A geometric view

    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!
  49. L

    A Curvature of Flat Lorentz manifolds

    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...
  50. A

    I Ricci tensor for Schwarzschild metric

    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...
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