Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always exists (under the above hypothesis) a metric (e.g. the one induced by the inner product, as noted by @Thomas Rot) such that $H$ becomes a complete Riemannian Hilbert manifold (see, for instance, Biliotti L., Mercuri F. (2017) Riemannian Hilbert Manifolds. In: Suh Y., Ohnita Y., Zhou J., Kim B., Lee H. (eds) Hermitian–Grassmannian Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 203. Springer, Singapore). Moreover, we know that Hopf-Rinow Theorem does not hold in general under our assumptions, because $H$ is infinite dimensional. However, we do know that almost-geodesics always exist (see I. Ekeland: The Hopf-Rinow Theorem in infinite dimension. J. Diff. Geometry 13, 287-301 (1978)). Now, recall that a submanifold $M$ of $H$ is said to be totally geodesic if every geodesic in $M$ is also a geodesic in $H$. My question is related to a converse of this notion, which involves almost-geodesics instead of geodesics: under what conditions can we say that an almost-geodesic in $H$ is still an almost-geodesic in $M$, for some infinite dimensional submanifold $M$? Is there any known term for such manifolds?

EDIT: an almost-geodesic, following the above paper by Ekeland, is a $C^{\infty}$ path $c(t)$ from some point $c(0)=p \in H$ to some other point $c(1)=q \in H$ such that $$ \int_{0}^{1} \| \dot{c}(t) \|_{c(t)} dt \leq \epsilon + d(p,q)$$ for some $\epsilon > 0$. Theorem A in the cited paper by Ekeland states that:

Let $N$ be a complete infinite dimensional Riemannian manifold, and let $p,q \in N$. $\forall \epsilon >0$, there exists some $C^{\infty}$ path $c(t)$ from $p$ to $q$ such that $$ \int_{0}^{1} \| \dot{c}(t) \|_{c(t)} dt \leq \epsilon + d(p,q)$$ that is, every such path is an almost geodesic.

Here, with 'submanifold' I mean an embedded submanifold.

EDIT: as @Thomas Rot pointed out in the comments, the previous question can be trivially answered. As he observed, an interesting question then is: under what conditions is a manifold $\epsilon$-geodesic?

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    $\begingroup$ About your introduction: $H$ with the Riemannian metric induced by the inner product is itself is a complete Hilbert manifold right? Do you mean to study other complete metrics? $\endgroup$ – Thomas Rot Jun 30 '20 at 9:41
  • $\begingroup$ Yes, you are right, thanks for noting this. Actually, what I'm most interested in now is not the metric used (we just need one for which $H$ is complete, so that almost geodesics certainly exist), but the existence of almost geodesics on $H$ which are still almost geodesics on $M$. Any metric for which this happens will suffice, so we can consider the inner product $\endgroup$ – Manuel Norman Jun 30 '20 at 9:44
  • $\begingroup$ can you define the notion of "almost geodesic" for me? $\endgroup$ – Thomas Rot Jun 30 '20 at 9:52
  • $\begingroup$ Sure, I will add the definition in the question $\endgroup$ – Manuel Norman Jun 30 '20 at 9:52
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    $\begingroup$ There are mathematical questions here if you change the quantifiers. For example one can define an $\epsilon$-almost geodesic as a curve that satisfies the bound with the given $\epsilon$. The question is then what submanifolds are $\epsilon$-almost geodesic? Maybe you will need to add some extra conditions on how to make the measurment precise for long geodesics. I would not know the answer (even in the finite dimensional case). $\endgroup$ – Thomas Rot Jun 30 '20 at 13:30

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