Intuition for the Drift Term of the Laplace-Beltrami Operator

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).

• I am not sure this is what you are looking for but I don't see this $\mu^l$ as a drift term. The thing is that $\mu^l$ and $g^{ij}\partial_{ij}$ are not well defined (they depend on your particular coordinate system) while the laplace beltrami operator $g^{ij}\partial_{ij}+\mu^l\partial_l$ is a geometric object. – Thomas Richard Jul 3 '17 at 21:10
• @ThomasRichard Yes indeed, I agree with you; in fact, it is one of the things confusing me (how can it be a "real drift" if it always disappears locally in normal coordinates?) as mentioned in my original question on MSE. But, in fact, the quantity turns up in the original papers by Ito himself on this topic (pg. 2), as the "drift coefficient" (in modern terminology) of the SDE for Brownian motion. – user3658307 Jul 4 '17 at 0:06

You really should think of $\Delta$ as an $L^2$ self-adjoint, elliptic operator in it's own right irrespective of coordinates. $-\Delta$ has positive spectrum, with countable eigenvalues accumulating at $\infty$ etc. It's the average of the Hessian $\nabla^2 f$ in the sense that contracting the $(0, 2)$ tensor $\nabla^2 f$ with the metric $g$ produces a $(1, 1)$ tensor that you can trace (which is an averaging operation). In this sense there is no drift term.

On the other hand, there is a way think of the extra term $\mu$ coming from the connection. For a smooth function $f : M \to \mathbb{R}$, from the smooth structure alone, you can always form the differential $Df \in \Gamma^{\infty}(T^{\ast} M)$ as smooth section of the cotangent bundle. Without a connection, you can think of $Df : TM \to \mathbb{R}$ as a map between smooth manifolds, differentiate again, and this gives $$D^2 f : TTM \to \mathbb{R}$$

I used $D$ for the differential so as not to confuse this with the exterior derivative $d$ which would have $d^2 = 0$.

Now if you have a connection, you can split $TTM$ as $$TTM \simeq VTM \oplus HTM$$ into vertical and horizontal sub-bundles. In this particular case both $VTM$ and $HTM$ are isomorphic to $TM$. $VTM$ is the kernel of the map $d\pi : TTM \to TM$ where $\pi: TM \to M$ is the bundle projection and this is isomorphic to $TM$. The connection allows you to "split" the map $d\pi : TTM \to TM$ obtaining an injective bundle morphism $TM \to TTM$ complementary to $VTM \simeq TM$ whose image I'll denoted $HTM$. The vertical bundle consists of elements of $TTM$ that are tangent to the fibres of $TM$ while the horizontal bundles consists of elements of $TTM$ that are tangent to the base $M$.

To help clarify, for a general vector bundle $\pi : E \to M$, the kernel $VE$ of $d\pi : TE \to TM$ is isomorphic to $E$ and a connection gives a splitting $TM \to TE$ whose image is denoted $HE$ and such that $TE \simeq VE \oplus HE \simeq E \oplus TM$.

Now, what this all has to do with "drift" is that under the identification $TTM \simeq TM \oplus TM$, the $g^{ij} \partial_{ij}$ term comes from $VE$ - it's the term tangent to the fibre and corresponds to $\partial_{ij} f = D^2 f$. The other term - the one with the connection coefficients $\Gamma$ corresponds to the part that is tangent to the base $M$.

In other words, the $g^{ij} \partial_{ij}$ part arises from differentiating while moving along the fibres of $TM$ and the $\Gamma$ part arises from moving along the base $M$.

Thus the drift is measuring the change in the fibres of the bundle $TM$ as measured by the connection as the basepoint $x \in M$ varies.

The usual way to think about the drift term is as the trace of the torsion tensor of the manifold. This interpretation is worked out, with analogies in hydrodynamics, by Diego Rapaport in [1] and [2].

• Thank you for your answer! I am confused though. I thought the torsion vanishes for the Levi-Civita connection. But the drift term seems to be perfectly existent when using the Christoffel symbols as they are usually defined. Doesn't your answer imply the drift should vanish whenever the Levi-Civita connection is used? – user3658307 Jul 3 '17 at 23:28
• If I look at equations 1-4 of Ref. 2, I see that the torsion tensor $T^\nu_{\nu\beta}dx^\beta$ that appears in the Laplacian operator has a "cotorsion" contribution does not vanish for the Levi-Civita connection. – Carlo Beenakker Jul 4 '17 at 8:54