### Jacobi Manifolds

#### Posted by John Baez

Here at the conference Foundations of Geometric Structures of Information 2019, Aïssa Wade of Penn State gave a talk about Jacobi manifolds. She got my attention with these words: “Poisson geometry is a good framework for classical mechanics, while contact geometry is the right framework for classical thermodynamics. Jacobi manifolds are a natural bridge between these.”

So what’s a Jacobi manifold?

It’s really simple: a **Jacobi manifold** is a smooth manifold $M$ such that the vector space $C^\infty(M)$ is equipped with the structure of a Lie algebra

$\{\cdot, \cdot \} \colon C^\infty(M) \times C^\infty(M) \to C^\infty(M)$

and the bracket is ‘local’ in the following sense:

$\mathrm{supp} \{f,g\} \subseteq \mathrm{supp} f \cap \mathrm{supp} g$

The most famous Jacobi manifolds are the **Poisson manifolds**, where the Lie bracket obeys this extra rule:

$\{f,g h\} = \{f,g\} h + g \{f, h \}$

For *any* Jacobi manifold, the bracket can be written as

$\{f, g \} = (f d g - g d f)(v) + (d f \wedge d g)(\Pi)$

for some unique vector field $v$ and bivector field $\Pi$. $v$ and $\Pi$ need to obey some identities to ensure that the bracket obeys the Jacobi identity. If we’ve got a Jacobi manifold with $v = 0$, so that

$\{f, g \} = (d f \wedge d g)(\Pi)$

then our Jacobi manifold is a Poisson manifold, and $\Pi$ is called the **Poisson bivector** or **Poisson tensor**. Conversely, any Poisson manifold has

$\{f, g \} = (d f \wedge d g)(\Pi)$

for some bivector field $\Pi$.

So, generalizing from Poisson manifolds to Jacobi manifolds amounts to allowing a nonzero vector field $v$ in our formula for the bracket.

But Aïssa favors another way of thinking about Jacobi manifolds. Apparently a Jacobi manifold structure on $M$ gives a way of making some principal $\mathbb{R}^\ast$-bundle over $M$ into a Poisson manifold, where $\mathbb{R}^\ast$ is the multiplicative group of the reals. The Poisson structures we get this way are homogeneous of degree $-1$ with respect to the $\mathbb{R}^\ast$ action. I’m not sure I’ve got all the details right here, but she stated an equivalence of categories between Jacobi manifolds and something like “manifolds equipped with a principal $\mathbb{R}^\ast$ bundle equipped with a Poisson bracket that’s homogeneous of degree $-1$”.

This viewpoint sets up the connection to contact geometry. I understand how contact geometry is the right geometry for classical thermodynamics, and I understand how Poisson geometry is the right geometry for classical mechanics… so I should be almost ready to understand how Jacobi manifolds are a nice home for both these branches of physics!

But my insight quits right around here, so if you want more, I’m afraid you’ll have to try these papers, and the references therein:

Charles-Michel Marle, Jacobi manifolds and Jacobi bundles.

Luca Vitagliano and Aïssa Wade, Holomorphic Jacobi manifolds and complex contact groupoids.

By the way, Vitagliano and Wade are doing something a lot deeper than the simple stuff I’m discussing here. They are talking about *holomorphic* Jacobi manifolds, and answering the question “what is the global object whose infinitesimal counterpart is a holomorphic Jacobi manifold?” To understand this question you have to first realize that just as Lie algebras are infinitesimal versions of Lie groups, Poisson manifolds can be seen as infinitesimal versions of ‘symplectic groupoids’. This is a fascinating story! We can then generalize this idea to Jacobi manifolds… and then adapt it to holomorphic Jacobi manifolds. But I’m still interested in much more basic stuff, like: what are we doing if we use a Jacobi manifold rather than a Poisson manifold as the ‘phase space’ for a classical system?

## Re: Jacobi Manifolds

1) If one has a one-parameter family of noncommutative algebras, which at t=0 is commutative, by differentiating in t that commutative algebra gets a Poisson structure. When we extend this R^* action beyond t=0 to act on the whole family, should it scale t or leave it alone?

2) You’d probably have been interested in this talk http://www.northeastern.edu/tzhou/bhmn/Zhang19.html that just happened in the big Boston colloquium. (I wasn’t there, just saw the abstract.)