What is Differential geometry: Definition and 409 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. T

    I Rewriting Equation of Motion in terms of Dual Fields (Chern-Simons)

    I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where...
  2. K

    Differential geometry of singular spaces

    TL;DR Summary: Reference request Hello! Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/books/differential-geometry-of-singular-spaces-and-reduction-of-symmetry/7D73498C35A5975594605428DA8F9267 I found that...
  3. cianfa72

    I The Road to Reality - exercise on scalar product

    Hi, I'm keep studying The Road to Reality book from R. Penrose. In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...
  4. T

    I Dirac delta function in 2d polar coordinates

    In 3 d spherical coordinates we know that $$\triangledown \cdot \frac{\hat{\textbf{r}}}{r^2}=4π\delta^3(\textbf{r})$$ Integration over all## R^3## is 4π So when we remove the third dimensions and enter 2d polar coordinates then $$\triangledown \cdot...
  5. cianfa72

    I Differential operator vs one-form (covector field)

    Hi, I'd like to ask for clarification about the definition of differential of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##. As far as I know, the differential of a scalar function ##f## can be understood as: a linear map ##df()##...
  6. cianfa72

    I Integral curves of (timelike) smooth vector field

    Hi, suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it). Can its integral curves cross at any point ? Thanks. Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...
  7. J

    I Transformation of a sphere

    Any given sphere surface consists of a finite number of fixed points. If all these points on the surface were to rotate/flip in their locations by 180° in respect to the centre of the sphere simultaneously and hence making the entire sphere turn outside in, how do you go about formulating this...
  8. G

    I Trouble with metric. Holonomic basis and the normalised basis

    ##df=\frac {\partial f}{\partial r} dr+\frac {\partial f}{\partial \theta}d\theta\quad \nabla f=\frac{\partial f}{\partial r}\vec{e_r} +\frac{1}{r}\frac{\partial f}{\partial \theta }\vec{e_\theta }## On the other hand ## g_{rr}=1\:g_{r\theta}=0\:g_{\theta r}=0\;g_{\theta\theta}=r^2\;##So...
  9. binbagsss

    A Tensor/Vector decomposition/representation & DOF arguement

    In fluid mechanics, it is sometimes useful to present the velocity, ##U## in terms of a scalar potential ##\Phi## as: ##\vec{U}=\nabla \phi## ##U## has 3 dof. ##\phi## has 1. If asked why this works, in terms of a dof argument, why is this? e.g . compared to GR common decomposition of the...
  10. AndreasC

    I What does Riemann mean?

    I was reading Bernhardt Riemann's old foundational text on abelian functions, and I found a part that really confused me. What he is trying to do is set up an invariant to classify 2d surfaces as simply connected, multiply connected, etc via some kind of "connectivity number". From the text, I...
  11. binbagsss

    I First algebraic Bianchi identity of Riemann tensor (cyclic relation)

    I am guessing that: $R_{a[bcd]}=0$ can not be derived from the symmetries of $R_{ab(cd)]}=R_{(ab)cd}=0$ $R_{[ab][cd]}=0$ ? Sorry when I search the proof for it I can not find much, it tends to come up with the covariant Bianchi instead. I am guessing it will need one of the symmetries...
  12. rajsekharnath

    Classical Source recommendation on Differential Geometry

    I am intending to join an undergrad course in physics(actually it is an integrated masters course equivalent to bs+ms) in 1-1.5 months. The thing is, in order to take a dive into more advanced stuff during my course, I am currently studying some of the stuff that will be taught in the first...
  13. sarriiss

    A Preserving Covariant Derivatives of Null Vectors Under Variation

    Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b},\\ \nabla_{a}n^{a}=0,\\ \nabla_{a}l_{b}=-\kappa n_{a}l_{b},\\ \nabla_{a}l^{a}=\kappa$$. How to show that under the variation of the null...
  14. Baela

    A Infinitesimal Coordinate Transformation and Lie Derivative

    I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
  15. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  16. G

    A Principal Invariants of the Weyl Tensor

    It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of...
  17. D

    I Are the coordinate axes a 1d- or 2d-differentiable manifold?

    Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold? Thank you!
  18. D

    I Is the projective space a smooth manifold?

    Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math
  19. malawi_glenn

    Other Collection of Free Online Math Books and Lecture Notes (part 1)

    School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...
  20. Introduction/Logic of propositions and predicates- 01 - Frederic Schuller

    Introduction/Logic of propositions and predicates- 01 - Frederic Schuller

    This is from a series of lectures - "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller
  21. S

    Normal vector of an embedding surface

    I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...
  22. diffgeo4life

    I Generic Curve in R^n: What We Know

    What do we know of a curve(/what can it look like) in R^n if we know that κ1,κ2,...,κn-1 is constant?
  23. V

    I Diverging Gaussian curvature and (non) simply connected regions

    Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...
  24. Somaiyah

    Help with deciding electives: Differential Geometry or Quantum Info

    Hello everyone, I wanted some help deciding which elective to choose. I am a junior and for my next semester I have the option to pick either Differential Geometry-I or Quantum Information. I am confused which one to choose. We will be doing QMII as a compulsory course next semester and I have...
  25. cianfa72

    I Darboux theorem for symplectic manifold

    Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
  26. P

    A Representing flux tubes as a pair of level surfaces in R^3

    I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of the functions represent flux lines. I am trying to solve this problem in ## R^3 ## with a...
  27. Falgun

    Geometry Confusion about Differential Geometry Books

    I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books. Among the others these two books by the same author seem to be the most widely recommended: Elementary Differential Geometry (Barret O' Neill) Semi-Riemannian Geometry with...
  28. I

    Geometry Geometrical books (differential geometry, tensors, variational mech.)

    I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of...
  29. T

    A Differential Geometry Class: Suggestions Welcome

    Can anyone recommend a good on-line class for differential geometry? I'd like to start studying GR but want a good background in differential geometry before doing so. Many thanks.
  30. T

    A Hypersurface Definition Confusion in General Relativity

    In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function can't be equal to a constant and its partial differential be...
  31. D

    I Commutative algebra and differential geometry

    In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?
  32. B

    Find the osculating plane and the curvature

    I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...
  33. O

    A On the relationship between Chern number and zeros of a section

    Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...
  34. steve1763

    A Find 2D Geometry of Line Element in Coordinates

    i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element 𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙 where 0≤𝜙<2𝜋 and 0≤𝜃<𝜋/2 Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...
  35. V

    I Riemannian Fisher-Rao metric and orthogonal parameter space

    Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
  36. K

    A Can we always rewrite a Tensor as a differential form?

    I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...
  37. K

    Geometry Modern Differential Geometry Textbook Recommendation

    Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff...
  38. K

    A Differential Forms or Tensors for Theoretical Physics Today

    There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
  39. Ishika_96_sparkles

    I Directional Derivatives of a vector ----gradient of f(P)----

    Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector $$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...
  40. W

    I Maximally Symmetric 3-Spaces

    Why does the constraint: $$R_{ijkl}=K(g_{ik} g_{jl} - g_{il}g_{jk})$$ Imply that the resulting space is maximally symmetric? The GR book I'm using takes this relation more or less as a definition, what is the idea behind here?
  41. Adrian555

    A Geodesics of the 2-sphere in terms of the arc length

    I'm trying to evaluate the arc length between two points on a 2-sphere. The geodesic equation of a 2-sphere is: $$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$ According to this article: http://vixra.org/pdf/1404.0016v1.pdf the arc length parameterization of the...
  42. 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?
  43. 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\\...
  44. 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\\...
  45. 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}##...
  46. 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...
  47. 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.
  48. 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...
  49. 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...
  50. 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...