Direction of the gradient of a monotone function on the sphere

Question

Let $\mathbb{S}^{n-1}$ denote the unit sphere in $\mathbb{R}^{n}$. Suppose $f:\mathbb{S}^{n-1}\rightarrow \mathbb{R}$ is increasing along a certain circle (I am interested in great circles in particular) that lies on $\mathbb{S}^{n-1}$. My intuition is that the surface gradient $\nabla_{\mathbb{S}^{n-1}} f$ is tangent to that circle. Correct ?

Answer 1

The gradient of a function (when you have a metric defined, like here) is a vector tangent to the surface of $S^{n-1}$. But is not necessarily tangent to any circle along which the function increase. In fact, it will be tangent only to the curve of maximal increase.