Surface curves equidistant from a simple closed geodesic Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is oriented, it partitions $S$ into two "halves," $S^+$ and $S^-$.
I am interested in learning properties of the curves that are equidistant from $\gamma$.
Define
$$\gamma^+(\delta) = \{ x \in S^+ \;|\; d(x, \gamma) = \delta \} \;,$$
where $d(x,A)$ is the length of the shortest path on $S$ from $x$ to any point in set $A$.

Q1. Does $\gamma^+(\delta)$ have a name in the literature? Has it been studied?
Q2. Under what conditions is $\gamma^+(\delta)$ a geodesic?
  Presumably rather stringent conditions on $S$. 
Q3. Under what conditions is $\gamma^+(\delta)$ a simple, closed curve?
  It might partition into several disconnected components. But perhaps for convex $S$,
  it is always a simple, closed curve?
Q4. Under what conditions could $\gamma^+(\delta)$ be a single point?
  [Revised to reflect Will Jagy's comment.]
  (Analogous to the north pole with respect to the equator on a (geometric) sphere.)

Thanks for pointers and help!

(Image below suggestive only!)

 
 
 
 
 
 
 
 
 

 A: To address question 2, consider perpendicular geodesics connecting $\gamma$ with $\gamma^+(\delta)$. These must be perpendicular with $\gamma^+(\delta)$, too, or else moving slightly in one direction or the other will decrease the distance below $\delta$. Consider the quadrilateral formed by a small piece of $\gamma$, perpendicular geodesics, and a piece of $\gamma^+(\delta)$. The Gauss Bonnet theorem implies that the Gaussian curvature of the interior equals the curvature of the piece of $\gamma^+(\delta)$. If the piece of $\gamma^+(\delta)$ is a geodesic its curvature is $0$, so the total Gaussian curvature in the quadrilateral must be $0$. 
The Gaussian curvature doesn't have to be identically $0$ between $\gamma$ and $\gamma^+(\delta)$. For example, you can revolve $y=2+\sin x$ about the $x$-axis and there are parallel geodesics where $x$ is a multiple of $\pi$.
(Image added by J.O'Rourke.)
 
 
 
 


Along any geodesic perpendicular to $\gamma$ at $v$ we can construct a function of the Gaussian curvature at distance $d$. Here is a construction that shows the function can change with $v$, unlike the surfaces of revolution: Choose a cylinder $\mathbb{R} \times C$ where $C$ is some plane curve. Choose a smooth function $f$, let $S$ be the surface of points $(x,y,z)$ which are of distance $f(x)$ away from the cylinder. If $f$ has a critical point at $x_0$ then the points of $S$ where $x=x_0$ form a geodesic. We can choose $C$ so that it has flat parts and curved parts, and then the Gaussian curvature does not depend only on $x$ because it is $0$ parallel to the flat parts of $C$. 
