differential geometry Definition and Topics - 167 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. 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...
  2. 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...
  3. T

    A Question about definition of a hypersurface

    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 cant be equal to a constant and its partial differential be...
  4. 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...
  5. 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...
  6. steve1763

    A Line element geometry

    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. Ive tried converting the coordinates to cartesian to see if it yields a...
  7. 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}...
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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?
  13. Adrian555

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

    I'm trying to evaluate the arc lenght 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 lenght parameterization of the...
  14. 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\\...
  15. 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\\...
  16. 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}##...
  17. Celso

    I Curve inside a sphere

    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.
  18. abby11

    A Derivation of radial momentum equation 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...
  19. 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...
  20. L

    A BMS coordinates near future 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...
  21. 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...
  22. 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...
  23. 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}##...
  24. Abhishek11235

    Differential 1 form on line

    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...
  25. Zhang Bei

    I Is the commutator of vector fields an important notion?

    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!
  26. 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}')...
  27. 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...
  28. 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...
  29. 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
  30. 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...