Please pardon, I wasn't sure how to phrase this to get an answer on Google. I'm familiar with the geodesic equation for a Riemannian Manifold which generalizes the concept of a straight line to this space. I was wondering if such an equation exists for circles?
I had some ideas.
The length between two points is $$\delta L=\int_a^b\sqrt{g_{\alpha\beta}\,dx^\alpha\,dx^\beta}$$ given the constraint, $$\ddot{x}^k=-\Gamma_{ij}^k\dot{x}^i\dot{x}^j.$$ .
Leave a constant and adjust b, preserving the length, we should draw out a circle. The Fundamental Theorem of Calculus might come into play.
Alternatively, the gradient of $$\nabla\delta L\cdot{d\vec{p}}=0$$ might yield a useful path, but I'm not quite sure how to properly differentiate this under the integral.
My intuition suggests I can assume any direction perpendicular to the tangent of the curve at the point on the circle will suffice, but I have some gaps in a vague proof in my head.
Given an equation for a a surface, the cross product of the normal with the tangent of the least distance curve might give a valid direction to goo in.
$$\frac{d\vec{p}}{d\lambda}=\frac{d\vec{s}}{d\lambda}\times\nabla{f}$$ where f is a constraint defining the surface, and $$\frac{d\vec{s}}{d\lambda}$$ is the tangent of the geodesic at b?
That's an interesting question. I don't think you could properly talk about a "circle" in a Riemannian manifold, but you could certainly consider curves with constant curvature as a natural generalization. I'll let you figure what the equation for that is in coordinates, if that's what you want. Intrinsically, you can define the curvature of a curve $\gamma(t)$ as:
$$\kappa(t) = \left\Vert \frac{D}{dt} T(t)\right\Vert$$ where $T(t) = \frac{\gamma'(t)}{\Vert \gamma'(t)\Vert}$ is the unit tangent vector to the curve and $\frac{D}{dt}$ is the covariant derivative along $\gamma$.
On the other hand, another generalization of "circle" would be a curve all of whose points are at a constant geodesic distance from some point (the "center").
In general, these two notions of "generalized circle" do not agree.