# About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves

In the paper Morse theory on Hilbert manifolds'' (1963), on page 326, Richard Palais makes a remark that if $\phi\colon V \to W$ is an isometry (of submanifolds of $\mathbb{R}^n$), then this does not imply that the natural lift $\bar{\phi}\colon H_1([0,1];V) \to H_1([0,1];W)$ is an isometry of Hilbert manifolds. There is no explanation of why this is not generally true, though. This remark is puzzling me, since I don't see how this would fail to be true.

A little bit of background details. Palais constructs the Hilbert manifold $H_1([0,1];V)$ non-intrinsically (as compared to the construction in Klingenberg's Riemannian geometry'') through the embedding of $V$ into some $\mathbb{R}^n$ and then choosing a metric on $\mathbb{R}^n$ such that $V$ is a totally geodesic submanifold.

So my question is: what is wrong with my argument below that $\bar{\phi}$ is indeed an isometry?

Let $V,W$ be two embedded submanifolds in Euclidean space, so $V,W$ become Riemannian manifolds with metric given by the restriction of the Euclidean metric to their respective tangent bundles. Let $\phi\colon V \to W$ be an isometry.

Now let $\sigma \in H_1([0,1];V)$ and $\lambda,\mu \in H_1([0,1];V)_\sigma$. The metric $g$ on $H_1([0,1];V)$ is defined by $$g_\sigma(\lambda,\mu) = \int_0^1 \langle \lambda(t), \mu(t) \rangle \,dt$$ with $\langle \,\cdot\,, \,\cdot\, \rangle$ the Euclidean inner product. Thus, with $\bar{\phi}(\sigma) = \phi \circ \sigma$ we get (using $d\bar{\phi}$ as given in Theorem~7 of the paper) $$\bar{\phi}_*(g)_{\bar{\phi}(\sigma)}(d\bar{\phi}_\sigma(\lambda), d\bar{\phi}_\sigma(\mu)) = \int_0^1 \langle d\phi_{\sigma(t)}(\lambda(t)), d\phi_{\sigma(t)}(\mu(t)) \rangle \,dt.$$ Since $\phi$ was an isometry, this reduces to the metric on $H_1([0,1];W)$.

Note that I interpreted $H_1([0,1];V)$ and $H_1([0,1];W)$ as Riemannian manifolds, where I assumed that the metric is defined by the inner product on the canonical Hilbert spaces on which the tangent spaces are modeled.

• it seems that Palais meant something else, or made a mistake – Misha Verbitsky Sep 26 '13 at 9:26
• The problem lies in the line above defining $g_\sigma(λ,μ)$. This is the metric for $H_0$, NOT the metric for $H_1$. The correct definition of the metric for $H_1$ uses $⟨λ'(t),μ'(t)⟩$ rather than $⟨λ(t),μ(t)⟩$ (see page 222 of my article where both metrics are defined), and with this change it is fairly obvious why my remark in the cited paper is in fact correct. – Dick Palais Sep 30 '13 at 3:20
• @DickPalais: indeed, thank you for pointing this out. My understanding now is that the derivatives of $\lambda,\mu$ are non-intrinsically defined using the (simple Euclidean) covariant derivative of the ambient space of the embedding. – Jaap Eldering Sep 30 '13 at 8:59