# Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

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?

• 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? Jun 30 '20 at 9:41
• 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 Jun 30 '20 at 9:44
• can you define the notion of "almost geodesic" for me? Jun 30 '20 at 9:52
• Sure, I will add the definition in the question Jun 30 '20 at 9:52
• 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). Jun 30 '20 at 13:30