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Peter Michor
Peter Michor
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Edit. Some more remarks:

If you have a strong Riemannian metric (such that $g_x$ induces the topology on $T_xM$ for each $x$), then you have a Hilbert manifold, the Riemannian exponential mapping is a local diffeomorphism, and by the Gauss lemma geodesic distance describes the manifold topology.

If the Riemannian metric is weak (so $g_x: T_xM\to T_x^*M$ is injective only), Then:
(1) one has to prove that the connection exists and is smooth.
(2) Even for a Banach manifold where the Riemannanian exponential mapping is automatically a local diffeomorphism, the Gauss lemma is not true in general, since the exponential mapping is a diffeomorphism on a neighborhood of $0\in T_xM$, but this need not be a $\|\cdot\|_{g_x}$-ball.
This is what happens all the examples described in the papers above.

The first paper has an example (concentric spheres, towards the end) of an incomplete geodesic where the conjugate points are dense.

We believe, that vanishing geodesic distance is tied to the fact, that sectional curvature is locally positive unbounded: behind mountain you always find a shorter geodesic.

For weak Riemannian metrics of low Sobolev order we do not know whether the geodesic equation is well posed. Of course KdV and Burgers are, but Burgers' close relative, the $L^2$-metric on $\text{Imm}(S^1,\mathbb R^2)$ or $\text{Imm}(S^1,\mathbb R^2)/\text{Diff}(S^1)$, are not known to have well posed geodesic equation.
In the paper

  • arXiv:1202.5122 Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle. Joachim Escher (IFAM), Boris Kolev (LATP),

the geodesic equation for the Sobolev $1/2$-metric on $\text{Diff}(S^1)$ is shown to be well posed, but the third paper above shows that it has vanishing geodesic distance.

The OP question stats from a (Banach)-manifolds and asks for Riemannian metric on it. But even if you find one, there are many more obstacles until one ends up with a metric space described by geodesic distance.

In my view, the metric (or even the geodesic equation) is more important than the manifold, which you can adapt to the metric somewhat.

Edit. Some more remarks:

If you have a strong Riemannian metric (such that $g_x$ induces the topology on $T_xM$ for each $x$), then you have a Hilbert manifold, the Riemannian exponential mapping is a local diffeomorphism, and by the Gauss lemma geodesic distance describes the manifold topology.

If the Riemannian metric is weak (so $g_x: T_xM\to T_x^*M$ is injective only), Then:
(1) one has to prove that the connection exists and is smooth.
(2) Even for a Banach manifold where the Riemannanian exponential mapping is automatically a local diffeomorphism, the Gauss lemma is not true in general, since the exponential mapping is a diffeomorphism on a neighborhood of $0\in T_xM$, but this need not be a $\|\cdot\|_{g_x}$-ball.
This is what happens all the examples described in the papers above.

The first paper has an example (concentric spheres, towards the end) of an incomplete geodesic where the conjugate points are dense.

We believe, that vanishing geodesic distance is tied to the fact, that sectional curvature is locally positive unbounded: behind mountain you always find a shorter geodesic.

For weak Riemannian metrics of low Sobolev order we do not know whether the geodesic equation is well posed. Of course KdV and Burgers are, but Burgers' close relative, the $L^2$-metric on $\text{Imm}(S^1,\mathbb R^2)$ or $\text{Imm}(S^1,\mathbb R^2)/\text{Diff}(S^1)$, are not known to have well posed geodesic equation.
In the paper

  • arXiv:1202.5122 Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle. Joachim Escher (IFAM), Boris Kolev (LATP),

the geodesic equation for the Sobolev $1/2$-metric on $\text{Diff}(S^1)$ is shown to be well posed, but the third paper above shows that it has vanishing geodesic distance.

The OP question stats from a (Banach)-manifolds and asks for Riemannian metric on it. But even if you find one, there are many more obstacles until one ends up with a metric space described by geodesic distance.

In my view, the metric (or even the geodesic equation) is more important than the manifold, which you can adapt to the metric somewhat.

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Peter Michor
Peter Michor
  • 25.3k
  • 2
  • 64
  • 112

There are infinite dimensional weak Riemannian manifolds with vanishing geodesic distance (in the sense as defined in the question). These are modeled on nuclear Frechet spaces, but the results extend to Sobolev completions of high enough order ($>\dim/2 +2$). They are still weak Riemannian manifolds (i.e., the Riemann metric does not generate the topology on the tangent spaces).

The first example was the $L^2$ metric on $\text{Emb}(S^1,\mathbb R^2)/\text{Diff}(S^1)$, as shown in the first paper below. Then it turned out that the right invariant $L^2$-metric on each full diffeomorphism group also has this property, also Sobolev metrics for Sobolev order $<1/2$ ($\le 1/2$ on $\text{Diff}(S^1)$). In particular, Burgers' equation and KdV are nonlinear PDE's corresponding to geodesic equations for metrics with vanishing geodesic distance.

All the papers are in the arXiv or on my homepage.

  • Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48.

  • Peter W. Michor; David Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10 (2005), 217--245 (written later)

  • Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Glob. Anal. Geom. 41, 4 (2012) 461-472.

  • Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom. 44, 1 (2013), 5-21.

  • Martin Bauer, Martins Bruveris, Peter W. Michor: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. 7 pages. To appear in: Ann. Glob. Anal. Geom.