Does a continuous scalar field on a sphere have continuous loop of "isothermic antipodes" For a continuous scalar field on a circle, there is a diameter of the circle such that the endpoints of the diameter have the same value. If you think of the scalar field as "temperature", then what this says is that there are points on opposite sides of the circle that are the same temperature: isothermic antipodes.
So, for a continuous scalar field on a sphere, the same is true: there are isothermic antipodes. (Just consider any great circle.)
Now, is there more you can say? Can you say for example that there is a closed loop on the surface of the sphere such that every point on the loop has the same value as the other endpoint of its diameter?
 A: [edit: as predicted, there is a counterexample with no continuous loop. See addition below.]
I think that you may not quite get a loop, only a topological continuum (a compact connected subset) on the sphere whose complement contains multiple components.  The continuum can be gotten by elaborating Rahul's answer to choose a suitable component of the $g=0$ locus.
The existence of topologically wild continua such as the "Warsaw circle" suggests that you can draw such a creature on the sphere or projective plane and then extend to a continuous function that would give a counterexample.  Or you could take a field that has the equator as the locus of isothermal antipodes ($g=0$) and try to perform a (antisymmetric) bending construction that modifies parts of the equator, turning it into a wild curve that cannot be traced by a continuous loop.
[added:  the extension construction would work as follows.  Take two opposite points on the equator.  Join them with a wild continuum in one hemisphere, and the antipode of that continuum in the opposite hemisphere. Define $f(x)$ to be the distance to the wild thing, in one hemisphere, and the negative of distance to the wild thing, in the opposite hemisphere.   Hence $f(x) = -f(-x)$ on the whole sphere, and $f=0$ only on the wild construction that cannot be traversed continuously by a path.] 
A: For any continuous scalar field $f$ on the sphere, you can define a continuous scalar field $g$ as $g(p) = f(p) - f(\bar{p})$, where $\bar{p}$ is the antipodal point of $p$. Unless $g$ is zero everywhere, there is a point $q$ for which $g(p) > 0$ and $g(\bar{q}) = -g(q) < 0$. As $g$ is continuous, a closed contour separating $q$ from $\bar{q}$ exists on which $g$ is zero.
Edit: The last statement "seems obvious" to me, but I don't have enough knowledge of compact topological spaces or manifolds to make it rigorous. Perhaps a counterexample exists.
Edit 2: See T..'s answer for a sketch of a counterexample. Apparently I was thinking about the existence of a curve while the question asked for a loop?
A: As for what more you can say in higher dimensions, take a look at the Borsuk-Ulam Theorem.
