Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?



Geodesic on Banach Manifold
Sorry not a real answer but too long for a comment:


You can put a "weak Riemannian metric" on a Banach manifold. Weak in this context means that the Riemannian metric does not describe the topology of the tangent space but induces a coarser topology. 


Example: The $L^2$-metric on the space $C^\infty ([0,1], \mathbb{R}^n)$. The natural topology is a Frechet topology, so it is not a Hilbert space. However the $L^2$ metric (with respect to the Hilbert space inner product on $\mathbb{R}^n$ is flat. This is for example used in Shape analysis. See Bauer, Michor et al.: Constructing reparametrization invariant metrics on spaces of plane curves; [1](http://arxiv.org/abs/1207.5965). I guess sifting a bit through overview articles on Riemannian metrics for shape spaces should turn up some valuable information. Further, in these works you find also information on curvature and geodesic distances (one of the big problems in the shape space buisness). 
Note that these concepts actually do not need Banach manifolds, but then you need to care about learning a bit of infinite-dimensional calculus (on locally convex spaces).


In the paper cited you find also references to a paper by Ebin and Marsden (D. G. Ebin and J. Marsden. Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 1970.) where the $L^2$-metric is used in a different context. If I recall correctly, Ebin and Marsden use the $L^2$-metric on a genuin manifold (i.e. the Frechet manifold $C^\infty (M,M)$ for a compact manifold $M$, so this example lives on an honest (Frechet) manifold.


