Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines? 



Embedding Riemmanian Manifold Linearly
No this is not possible, for example let $M$ be such that there exists a compact geodesic $\gamma: [0,1]\rightarrow M$ (for example $M=S^n$). n this case if such a map $\phi:S^n\rightarrow V$ did exist (where $V$ is a linear space) then $\phi\circ\gamma: [0,1] \rightarrow V$ must also be compact, however a line in $V$ is not compact; a contraction. 


