How can I calculate exponential map for cylinder?

Question

I want to calculate the exp map and log map for cylinder. But as shown figure, I only know the geodesic equation of cylinder is helix. And I search it in some books and website like the google scholar and differential geommetry, etc. But I also can't find it.

And I known the exp map define of differential geommetry, But I can't use it for calculate the exp map and log map.

So How can I calculate the exp map and log map for cylinder. Or if you may have the equation of exp and log map for cylinder, or the paper, it is will helpful to me, and thank you very much!

Answer 1

Not an answer,only a demonstration of the exponential mapping.

• The green contours is the trail of the exponential mapping.

Clear["Global`*"];
U = 200; L = 500;
ani = Manipulate[
  Block[{r = 1/κ}, 
   ParametricPlot3D[{0, r, 0} + 
     r {Cos[θ], Sin[θ], 0} + {0, 0, 
      u}, {θ, -π/2 - L/2/r, -π/2 + L/2/r}, {u, -U/2, 
     U/2}, PlotPoints -> 100, MaxRecursion -> 4, 
    PerformanceGoal -> "Quality", 
    PlotRange -> {{-L/2, L/2}, {-L/π, L/π}, {-U, U}}, 
    Boxed -> False, Axes -> False, BoxRatios -> Automatic, 
    MeshFunctions -> {Function[{x, y, z, θ, u}, 
       Sqrt[(r (θ + π/2))^2 + u^2]], 
      Function[{x, y, z, θ, u}, 
       ArcTan[r (θ + π/2), u]]}, MeshStyle -> {Red, Green},
     PlotStyle -> Automatic, 
    Lighting -> "Accent"]], {κ, .00001, 2 π/L}]

Answer 2

The exponential map in euclidean space $$\mathbb R^3 = \mathbb R^2 \times \mathbb R : \ (x,y,z) \to \{ (\Re, \Im)(e^{\xi + i \theta}), u\} $$ has Jacobian $$\frac{\partial (x,y,z) }{\partial (\xi,\theta,u)} $$

   MatrixForm[(D[{E^\[Xi] Cos[\[Theta]], E^\[Xi] Sin[\[Theta]], u}, #] &) \@
        {\[Xi], \[Theta], u}]

$$\left( \begin{array}{ccc} e^{\xi } \cos (\theta ) & e^{\xi } \sin (\theta ) & 0 \\ -e^{\xi } \sin (\theta ) & e^{\xi } \cos (\theta ) & 0 \\ 0 & 0 & 1 \\ \end{array} \right)$$

    G  =   J . J\[Transpose] // Simplify

$$\left( \begin{array}{ccc} e^{2 \xi } & 0 & 0 \\ 0 & e^{2 \xi } & 0 \\ 0 & 0 & 1 \\ \end{array} \right)$$

For the geodesics we use the simplifying Euler-Lagrange mechanism. Compute the square of the coordinate velocity list with metric G

$$ L = \frac{1}{2} v . G . v = \frac{1}{2} \left(e^{2 \xi } \overset{.}{\theta }^2+e^{2 \xi } \overset{.}{\xi }^2+\overset{.}{u}^2\right)$$

The differential equations of the geodesics are the equations of motion with respect to geodesic length as arc parameter $$ \left(\partial_t \partial_{\overset{.}{u_k}} - \partial_{u_k} \right) \ L ==0 $$