Motivations for the study of dual connections

There may be many motivations to study dual (or maybe rather: adjoint) connections. Let me give one. The metric $g$ identifies a vector bundle $V$ and its dual $V^*$. Any connection $\nabla$ on $V$ induces one on $V^*$, and the pullback of this by $g$ is the dual connection. They agree if and only if $\nabla$ and $g$ are compatible.

Here is one application. If $\nabla$ is flat, this can be used to construct a characteristic class of $(V,\nabla)$, see Bismut-Lott. The main idea is that $\operatorname{tr}((\nabla^*-\nabla)^{2k+1})\in\Omega^{2k+1}(M)$ is closed, and its cohomology class is independent of $g$. These classes are global obstructions against a parallel metric with respect to $\nabla$. Note that only if $\nabla$ is flat, the local obstructions against such a metric vanishes.

One motivation for studying dual (also called conjugate) connections in this sense comes from the study of the equiaffine geometry of a hypersurface.

Let $i:\Sigma \to \mathbb{A}$ be a nondegenerate cooriented codimension one immersion in flat $(n+1)$-dimensional affine space $\mathbb{A}$. That the immersion be nondegenerate means that its second fundamental form (a normal bundle valued symmetric two tensor) is nondegenerate. That it be cooriented means that its normal bundle is orientable. The coorientation and the second fundamental form determine a (pseudo-Riemannian) conformal structure $[g]$ on $\Sigma$. A choice of parallel volume form $\Psi$ on $\mathbb{A}$ determines a transverse vector field known as the equiaffine normal distinguished by the following requirements. First, such a transversal $N$ determines two volume densities on $\Sigma$, one, $|\iota(N)\Psi|$ via interior multiplication with the fixed ambient volume form, and the other the volume element of the representative, $g$, of the second fundamental form corresponding to the transversal $N$, viewed as a pseudo-Riemannian metric. These are required to agree. The transversal also determines a torsion-free affine connection $\nabla$ on $\Sigma$, and this connection is required to preserve the volume density.

The oriented projectivization of the vector space $\mathbb{A}^{\ast}$ dual to $\mathbb{A}$ carries a flat projective structure. The conormal Gauss map associates to each $p \in \Sigma$ the parallel translate to the origin of the ray in $T_{i(p)}\mathbb{A}$ annihilating $T_{i(p)}i(\Sigma)$. The pullback of the flat projective structure on the oriented projectivization of $\mathbb{A}$ via the conormal Gauss map induces on $\Sigma$ a flat projective structure. The equiaffine normal determines a unique immersion from $\Sigma$ to $\mathbb{A}^{\ast}$ covering the conormal Gauss map; the pullback via this map of the flat affine connection on $\mathbb{A}^{\ast}$ dual to that on $\mathbb{A}$ is a representative $\bar{\nabla}$ of the induced flat projective structure, and it can be checked that $\bar{\nabla}$ is conjugate (dual) to the connection $\nabla$ decribed in the previous paragraph with respect to the metric $g$.

In general the connections $\nabla$ and $\bar{\nabla}$ are different. Precisely, they coincide if and only if both equal the Levi-Civita connection of $g$, in which case the image $i(\Sigma)$ is necessarily an open subset of a hyperquadric, by a theorem of Maschke-Pick-Berwald.

On the other hand, it is not true that an arbitrary pair of conjugate connections can be obtained in this way. In the pair of conjugate connections arising on a nondegenerate cooriented hypersurface immersion, at least one of the pair is necessarily projectively flat.

The preceding can mostly be found in some form in the textbook Affine Differential Geometry by Nomizu and Sasaki. See also section $6$ of the Survey on affine spheres by J. Loftin.