In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as: $$ \Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell $$ The second term: $$ \mu^\ell = - g^{jk}\Gamma^\ell_{jk} $$ can be viewed as the "convection term" in the Riemannian heat equation or the drift term of the (Ito) stochastic differential equation defining Brownian motion on $(M,g)$.

Question: What is the geometric meaning of this $b$?

I would really like some intuition as to how the geometry generates this term (i.e. how to interpret it geometrically). Any other insights into intuitively understanding this term would be appreciated as well (e.g. other places where it appears).

(Note: this is a refinement of this question).

In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as: $$ \Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell $$ The second term: $$ \mu^\ell = - g^{jk}\Gamma^\ell_{jk} $$ can be viewed as the "convection term" in the Riemannian heat equation or the drift term of the (Ito) stochastic differential equation defining Brownian motion on $(M,g)$.

Question: What is the geometric meaning of this $\mu$?

I would really like some intuition as to how the geometry generates this term (i.e. how to interpret it geometrically). Any other insights into intuitively understanding this term would be appreciated as well (e.g. other places where it appears).

(Note: this is a refinement of this question).