On generalized Tanaka connection

Question

Many authors used the Tanaka connection in their papers such as [1] to define new Tanaka connection so-called Generalized Tanaka connection $^*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\phi,g)$ as follows:

$$^*\Gamma_{ik}^j = \Gamma_{ik}^j + \eta_j\phi_k^i-\nabla_j\xi^i\eta_k+\xi^i\nabla_j\eta_k.$$

My main question: Why we need to define another new connection such as Tanaka connection even though we have Levi-Cevita connection?

[1]:Shukichi Tanno, MR 1000553 Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), no. 1, 349--379.

Answer 1

The short answer to your question is that the Levi-Civita connection is perfectly adapted to the metric, but not compatible with the additional structure around. In particular, it does not preserve the contact distribution (in your notation, this should be the kernel of $\phi$). Indeed, a torsion-free connection can never preserve a non-involutive subbundle of the tangent bundle.

In general, given some kind of geometric structure, you can look for connections compatible with the structure. If there are such connections (which is always the case if the structure can be described by a reduction of the linear frame bundle), then there may be a part of torsion that is independent of the connection ("intrinsic torsion"). You can then try to find a canonical connection by choosing a connection for which "the remaining torsion" vanishes, which means that torsion satisfies a "normalization condition" (which may have to be chosen). You can also check how many connections are there with the same torsion (always only one if your structure involves a pseudo-Riemannian metric). This whole process is formalized in the concept of Sternberg prolongation, and in favourable situations it leads to a canonical connection.

If you run this process for Riemannian metrics, you of course get the Levi-Civita connection, but for subclasses (like Hemitian metrics on almost complex manifolds, or contact metric structures), you get connections like the Tanaka connection. These are better adapted to the additional structure (to which the Levi-Civita connection is not adapted in general), but have non-trivial torsion.