Can linear connections other than Levi-Civita connections be useful? Consider a smooth Riemann manifold such that a Levi-Civita connection is defined.
I am wondering whether there are examples in mathematics or physics where the use of other linear connections is useful despite the fact that a Levi-Civita connection is available.
 A: In Ricci flow we deal with an evolving metric $g(t)$ on a manifold $M$, which can be thought of as a metric on the spatial tangent bundle $TM \times \mathbb R \subset T(M \times \Bbb R)$. 
One obvious way to put a geometry on this "spacetime manifold" $M\times \Bbb R$ is to define a metric $h(x,t) = g(x,t) + dt^2$, but the Levi-Civita connection $\nabla'$ that we get is not so intuitive: for example if $X,Y$ are time-constant spatial vector fields such that $\nabla_X Y = 0$ ($\nabla$ the Levi-Civita connection of the slice $(M,g(t))$) then $$\nabla'_X Y = -\frac12 (\partial_t g)(X,Y)\partial_t$$ is not necessarily zero - that is, the action of $\nabla'$ on spatial vector fields is not the same as that of $\nabla$.
By instead choosing the unique linear connection $D$ extending $\nabla$ such that $D \partial_t = 0$ and $g(D_t X, Y) = \frac12 \partial_t g(X,Y)$ for time-constant spatial $X,Y$, we get a connection that is very nice to work with for Ricci flow: not only does the fact that it extends the spatial connection make computations feel quite natural, but the evolution equation for the (spatial) curvature tensor reduces to a reaction-diffusion equation $$D_t R = \Delta R + Q$$ where $Q = R\#R + R^2$ is a well understood quadratic curvature term. This simplification (known as Uhlenbeck's trick) can also be achieved by choosing time-varying frames or a time-varying bundle map; but I think this geometric realization is the prettiest way to do it.
If you're interested in the details you can find them in chapter 5.3 of this book.
A: Absolutely yes. In Continuum Mechanics, a non-metric connection is used to describe uniform bodies, i.e., bodies that have intrinsically the same material properties at each point, but could be locally distorted, so that these properties appear to be dependent on the point. A beautiful reference is Epstein and Maugin (1990), The energy-momentum tensor and material uniformity in finite elasticity, Acta Mechanica 83, 127-133
