Suppose that I am interested in a probability measure $\mu$ defined on the sphere $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$, whose density with respect to the normalised surface measure can be written as $\exp(-V(x))$, with $V: \mathbb{R}^d \to \mathbb{R}$.
My question is: concretely, what matrices do I need to write down in order to verify that $\mu$ satisfies the Bakry-Émery criterion with a given constant $\rho \geqslant 0$?
Assume that I have little background in differential geometry, and don't yet want to subject myself to carefully understanding things like Ricci curvature etc. (perhaps another day!). I am fairly sure that this task reduces to explicitly computing a few derivatives of $V$ (which I insist on writing as a map with domain $\mathbb{R}^d$), and then verifying that a related matrix is positive semidefinite, and so I am looking for concrete instructions which will allow me to do so efficiently.
The criterion you have to check is $$ \mathrm{Hess}_x V[v,v]+\mathrm{Ric}_x(v,v)\geq \rho g_x(v,v) $$ for all $x\in S^{d-1}$ and unit vectors $v\in T_x S^{d-1}$.
In the case of the round metric on $S^{d-1}$, things get considerably simpler. First, $T_x S^{d-1}=\{x\}^\perp\subset \mathbb R^d$ and $g_x(v,v)=\lvert v\rvert^2$, the Euclidean norm squared. Moreover, the Ricci curvature is just $\mathrm{Ric}_x(v,v)=(d-1)g_x(v,v)$.
Finally, the Hessian can be computed as follows: If $\gamma$ is the (locally unique) geodesic with starting point $x$ and starting velocity $v$, then $$ \mathrm{Hess}_x V[v,v]=\frac{d^2}{dt^2}\bigg|_{t=0}V(\gamma_t). $$ In the case of the sphere, the geodesic is given by $\gamma_t=(\cos t)x+(\sin t)v$. More explicitly, this amounts to $$ \mathrm{Hess}_x V[v,v]=\langle v,\nabla^2 V(x)v\rangle-\langle x,\nabla V(x)\rangle, $$ where $\nabla V$ and $\nabla^2 V$ are the "Euclidean" gradient and Hessian of $V$ as function on $\mathbb R^d$.