I have stumbled upon the need to prove whether or not there is a unique closest distance between a point and a subset, as i understand this it is only provable given the subset is convex and thus i would like to prove or disprove this property.
The subset in question is
$$ \mathbb{M} = \{ \boldsymbol{\Psi}(\theta,\phi) \in \mathbb{C}^N \;: (\theta,\phi) \in \mathbb{R}^2 \} $$
where the generating map $\boldsymbol{\Psi}$ given in index notation is
$$ {\Psi}_j(\theta,\phi) = \gamma(\theta,\phi) e^{-i\langle \mathbf{k}(\theta,\phi), \mathbf{r}_j \rangle_{\mathbb{R}^3}} $$
where the $\langle , \rangle_X$ denotes the inner product in the space $X$, $\mathbf{r}_j$ are constant vectors in $\mathbb{R}^3$ and the function $\mathbf{k}$ maps the plane to the embedded 2-sphere i.e.
$$ \mathbf{k}(\theta,\phi) = \frac{2\pi}{\lambda} \begin{pmatrix} \cos(\theta)\sin(\phi) \\ \sin(\theta)\sin(\phi) \\ \cos(\phi) \end{pmatrix}. $$
I have gotten as far as calculating the metric tensor for the simple case where $\gamma(\theta,\phi) = 1$ giving
$$ g_{qp} = J_{kq}^*J_{kp} =\sum_{k=1}^N \frac{\partial \Psi^*_k}{\partial x_q} \frac{\partial \Psi_k}{\partial x_p} = \notag \\ = \sum_{k=1}^N \langle \frac{\partial\mathbf{k}(\theta,\phi)}{\partial x_q}, \mathbf{r}_k \rangle_{\mathbb{R}^3} \langle \frac{\partial\mathbf{k}(\theta,\phi)}{\partial x_p}, \mathbf{r}_k \rangle_{\mathbb{R}^3}. $$
where $J$ is the Jacobian. For my specific set of $\mathbf{r}$'s this turned out to be
$$ g= 375.7291 \begin{pmatrix} \sin^2(\phi) & 0 \\ 0 & \cos^2(\theta) \end{pmatrix} $$
But after this point i am at a loss, how to i proceed to prove/disprove the geodesic convexity of $\mathbb{M}$? I have a gut feeling that the Christoffel symbols of the second kind should be involved.
It should also be noted that i have not given any information on the function $\gamma$ but that it will, in the end, have a closed form expression and, although complicated, i can thus complete this with Maple or Mathematica once i understand the concept better.
Thanks in advance!