manifold

  1. J

    I How do charts on differentiable manifolds have derivatives without a metric?

    I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
  2. Avatrin

    A Logical foundations of smooth manifolds

    Hi I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...
  3. redtree

    I Fourier transform on manifolds

    Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
  4. A

    I Tangent vector basis and basis of coordinate chart

    I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...
  5. ZuperPosition

    A 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}')...
  6. E

    A Lie derivative of vector field defined through integral curv

    Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...
  7. K

    I Gradient vector without a metric

    Is it possible to introduce the concept of a gradient vector on a manifold without a metric?
  8. C

    I Injective immersion that is not a smooth embedding

    Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
  9. N

    A Chern-Simons Invariant

    I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold...
  10. C

    B Differentiable function - definition on a manifold

    Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...
  11. shahbaznihal

    A On metric and connection independence

    Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...
  12. darida

    A First Variation of Jacobi Operator

    <Moderator's note: Moved from a homework forum.> 1. Homework Statement From this paper. Let ##L## be the Jacobian operator of a two-sided compact surface embedded in a three-maniold ##(M,g)##, ##\Sigma \subset M##, and defined by $$L(t)=\Delta_{\Sigma(t)}+ \text{Ric}( ν_{t} , ν_{t}...
  13. S

    A A.continuation from a germ of the metric of complex manifold

    Suppose ##M## is a connected analytic manifold with metric ##g(x), x \in M## which is everywhere analytic. Define ##\gamma(g(x))## as the germ of the metric at the point ##x##. Question: Is it possible to come up with a nontrivial ##\gamma##, ##M_1##, and ##M_2##, where the germ ##\gamma##...
  14. DAirey

    I Does a metric exist for this surface?

    I have a surface defined by the quadratic relation:$$0=\phi^2t^4-x^2-y^2-z^2$$Where ##\phi## is a constant with units of ##km## ##s^{-2}##, ##t## is units of ##s## (time) and x, y and z are units of ##km## (space). The surface looks like this: Since the formula depends on the absolute value of...
  15. 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...
  16. R

    [Symplectic geometry] Show that a submanifold is Lagrangian

    1. 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...
  17. Heisenberg1993

    A Two cones connected at their vertices do not form a manifold

    Why is i that two cones connected at their vertices is not a manifold? I know that it has to do with the intersection point, but I don't know why. At that point, the manifold should look like R or R2?
  18. orion

    I Question about chart parameterization

    Suppose we have an n-dimensional manifold Mn and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝn. So far I'm clear on this. However, where I become confused is when some books say that φ-1 is called a parameterization of U and basically...
  19. S

    A Concept of duality for projective spaces and manifolds

    I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...
  20. G

    A Map from tangent space to manifold

    Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold M , for any vector k in the tangent space T_p at a point p\in M , we can find a path x^{\mu}(\lambda) that passes through p which corresponds to the geodesic for that...
  21. D

    Flow in network or manifold

    Hi, A chamber (manifold type cylinder) has 1 inlet and 8 outlets of 3 inch diameter each. 8 suction blowers are connected to the chamber's outlet. Each blower's suction flow rate is 1000 cfm. what will be the flow rate through the inlet of the chamber (diameter = 3 inches). Will the inlet...
  22. S

    Diffeomorphism and isometry

    Let f:p\mapsto f(p) be a diffeomorphism on a m dimensional manifold (M,g). In general this map doesn't preserve the length of a vector unless f is the isometry. g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V). Here, f_\ast:T_pM\to T_{f(p)}M is the induced map. In spite of this fact why...
  23. D

    Local parameterizations and coordinate charts

    I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them...
  24. D

    Coordinate charts and change of basis

    So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...
  25. D

    Differential map between tangent spaces

    I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
  26. D

    Integration on manifolds

    In all the notes that I've found on differential geometry, when they introduce integration on manifolds it is always done with top forms with little or no explanation as to why (or any intuition). From what I've manage to gleam from it, one has to use top forms to unambiguously define...
  27. D

    Why are vectors defined in terms of curves on manifolds

    What is the motivation for defining vectors in terms of equivalence classes of curves? Is it just that the definition is coordinate independent and that the differential operators arising from such a definition satisfy the axioms of a vector space and thus are suitable candidates for forming...
  28. D

    Attempting to understand diffeomorphisms

    I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts. Suppose that one has a...
  29. D

    Understanding the notion of a tangent bundle

    I've been reading up on the definition of a tangent bundle, partially with an aim of gaining a deeper understanding of the formulation of Lagrangian mechanics, and there are a few things that I'm a little unclear about. From what I've read the tangent bundle is defined as the disjoint union of...
  30. D

    Differentiability of a function on a manifold

    I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...
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