Given a linear function $f$ on euclidean space $\mathbb{R}^n$ we know that $f$ is of form $f(x) = b^{T}x$ and its gradient is constant.
My question of similar thing on riemannian manifolds. specifically, is there function $f: \mathcal{M} \rightarrow \mathbb{R}$ s.t riemannian gradient $\text{grad} f(x) = v$. Now this can't be possible, a tangent vector $v$ is not tangent vector at other points.
consider simple relaxation $\text{grad} f(x) = \text{PT}^{x_{0} \rightarrow x}(v) $ where $\text{PT}^{x_{0} \rightarrow x}$ is parallel transport from $x_{0}$ to $x$ along the minimizing geodesic (and assume such minimizing geodesic is unique). I am putting question for clarity below clarity
Is there are function $f: \mathcal{M} \rightarrow \mathbb{R}$ s.t $\text{grad} f(x) = \text{PT}^{x_{0} \rightarrow x}(v) $ where $v \in T_{x_{0}} \mathcal{M}$