The Sasaki metric is a canonical almost Hermitian metric on the tangent bundle of a Riemannian manifold (with an affine connection). It was originally discovered by Sasaki in 1958 and expanded on by Dombrowski in 1963. Since then, it has appeared in various different contexts and been studied in depth by many people. For details on its construction, there is a good paper by Hiroyasu Satoh with a lot of helpful computations.
Recently, it has shown up in some of our work on optimal transport. In particular, we found that for a certain type of cost function, the continuity of optimal transport depends on the geometry of a certain Kähler Sasaki metric.
However, instead of simply restating these results, I wanted to understand the geometry of these spaces a bit more. To do so, I thought it might be helpful to create some sketches. I'm hoping that these illustrations might be helpful for others who are trying to understand the geometry as well. For the explanations, I'm going to assume that the reader is familiar with some differential geometry. However, no background knowledge is necessary to play with the diagrams. I got some feedback that on Firefox some of the graphics weren't displaying, to get around this I have included a link to the files hosted on the Geogebra website.
What does the tangent bundle actually look like?
As a point of clarification, it's a bit of a misnomer to refer to a single Sasaki metric. In fact, it's a general construction that can be done on any Riemannian manifold with an affine connection (although it's only Kähler in the very specific case that the metric is Hessian).
In order to draw pictures, I'm going to focus on one particular example, which is the tangent bundle over a piece of a sphere. This is an important example for several reasons.
- It's the simplest example with interesting geometry.
- The example comes from statistics. For readers familiar with exponential families, the base metric is the so-called "probability simplex" and the round metric on the base is the Fisher metric. The flat connection that induces the Kähler complex structure is induced from the natural parameters of the trinomial distribution.
- This particular space that corresponds to one of the optimal transport problems we studied. In fact, this has applications to mathematical finance, as it can be used to show certain portfolio maps are smooth.
The Sasaki metric is defined on the tangent bundle, which corresponds to points in the sphere along with the possible directions that one could travel. In this example, to specify a point in the tangent space, we pick a blue base point and draw its tangent space, which corresponds to the plane tangent to the sphere. We then pick a gray point within this tangent space to specify a point and a tangent vector.
The Tangent Bundle:
Here, the tangent space is 4-dimensional, because there are two degrees of freedom for the blue point as well as two degrees of freedom for the gray point. However, this picture isn't that interesting, and the intrinsic geometry isn't at all apparent. Notice that this is not a complete manifold. The Sasaki metric often fails to be complete, so this is a prototypical example. It turns out that this example cannot be extended to any larger subset of the sphere, as well.
The Complex Structure
This tangent bundle is a Kähler manifold, it is worth trying it understand both its complex structure as well as its Riemannian metric. I have tried to depict both of these, starting with the complex structure (denoted J).
The basic idea in the following diagram is that from a red vector X, one computes -JX, which is the negative of it's image under the map J.
In this picture, I have chosen X to be a vertical vector (so-called because it corresponds to a direction in the tangent plane), whereas -JX is a horizontal vector, as it corresponds to moving in the base manifold (i.e. horizontally). Due to the fact that I'm drawing a 3-dimensional picture of something that is inherently 4-dimensional, some things are lost in translation. For instance, X and -JX are actually perpendicular with respect to the Sasaki metric. Note that X and -JX are tangent vectors to the tangent bundle (i.e. elements of TTM).
Note that the blue points are in the octant of the sphere, the red points are in the red plane and the green point is in the green plane. Points B,V, and X can all be manipulated, which will affect B' and -JX.
The Complex Structure:
For computational ease and to help with the visual intuition, I took a few artistic licenses with this diagram.
- To represent tangent vectors to the tangent bundle (i.e. elements of TTM), I used arrows between two points in the tangent bundle, which hopefully gets the idea across. In this way, at a point (B,V) in the tangent bundle, X is an arrow pointing to (B,V+X) whereas -JX is an arrow pointing to (B',V).
- Instead of trying to integrate out the exponential of X to find B', I just used a first order Taylor series in the natural parameters.
- In order to convey that the complex structure does not change the fiber direction, I orthogonally projected the vector from B to V into the tangent space of B' (i.e. from the red tangent space to the green) and then renormalized so that the lengths were preserved. As changing the base point stretches and skews the fibers, this isn't the most geometrically faithful representation, but I find it easier to picture.
The Riemannian Geometry
To try to explain how the fibers get distorted as the base point moves, I made the following picture. In terms of the Sasaki metric, the round metric on the sphere is a faithful representation of how the manifold "looks" while the grid corresponds to an orthonormal lattice on the fibers. This means the space is warped from our viewpoint in 3D, which isn't obvious if you just look at the first picture. Furthermore, as the base point moves, the warping changes in some complicated ways. As a disclaimer, there may be some errors in the precise lattice points. Nonetheless, I feel like it gets across the point of how the tangent spaces "change shape" depending on the base point.
The Riemannian geometry:
What about the curvature?
In our work, we related the curvature of this space to the regularity of a certain optimal transport problem in mathematical finance. However, I'm not sure of a good way to depict the curvature of this space, so that picture will have to wait.
I also don't have a convincing picture for how this connects to optimal transport. That's something I'll keep thinking about and will make more sketches if I figure something out.