On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.


Introduction
Geodesic distances naturally appear in the geometry of infinite dimensional manifolds.A new aspect is that they may also vanish on two distinct points.In general, the vanishing of the geodesic distance may occur for certain Riemannian metrics, where no special conditions are assumed, namely for weak Riemannian metrics, [1,Definition [5.2.12].These metrics are important, since they are the only possible metrics when the manifold is not modelled on a Hilbert space.
First examples of vanishing geodesic distances in infinite dimensional Fréchet manifolds were found in [4], [6] and [7].A simple example of vanishing geodesic distance can be also constructed in a Hilbert manifold, [5].However, one may still wonder whether replacing a weak Riemannian metric with a left invariant weak Riemannian metric with respect to a Hilbert Lie group structure somehow might give a condition to have positive geodesic distance on distinct points.
The answer to this question does not seem intuitively clear.For instance, we observe that connected, simply connected and commutative Banach Lie groups, equipped with a bi-invariant weak Riemannian metric have positive geodesic distance on distinct points.In short, their geodesic distance is actually a distance.The proof of this fact essentially follows from [9,Proposition IV.2.7], observing that the exponential mapping is a local Riemannian isometry.
Thus, the question is whether considering a left invariant weak Riemannian metric on a noncommutative, connected and simply connected Banach Lie group may prevent the vanishing of the geodesic distance.Our first result answers this question in the negative.
Theorem 1.1.There exists a left invariant weak Riemannian metric on the infinite dimensional Heisenberg group , whose associated geodesic distance is not positive on all couples of distinct points.
The Heisenberg group is modelled on the Hilbert space ℓ 2 × ℓ 2 × R, where ℓ 2 is the standard linear space of square-summable sequences.More details are given in Section 2. The same technique to prove the previous theorem also gives an analogous degenerate geodesic distance for the sub-Riemannian Heisenberg group.Theorem 1.2.There exists a left invariant weak sub-Riemannian metric on the infinite dimensional Heisenberg group such that its associated geodesic distance is not positive on all couples of distinct points.
The previous theorems are contained in Theorem 3.2 and their proof relies on the same sequence of length-minimizing curves.Furthermore, the proof of these results precisely shows that both Riemannian and sub-Riemannian distance are vanishing between points that have the same projection on the subspace ℓ 2 × ℓ 2 × {0}.Remark 3.1 completes the picture, showing that when the projections of two points on ℓ 2 × ℓ 2 × {0} are different, then both their Riemannian and sub-Riemannian distance are positive.
From another perspective, dealing with a left invariant weak Riemannian metric has the advantage to find the sectional curvature by more manageable formulas.In [6], Michor and Mumford proved that in different Fréchet manifolds with a vanishing geodesic distance the sectional curvature is unbounded.Theorem 1.3 below presents the same phenomenon for the left invariant weak Riemannian metric σ defined in (10), in the infinite dimensional Heisenberg group .
We wish to emphasize that for general weak Riemannian metrics the existence of the Levi-Civita connection is not guaranteed a priori, hence the same existence problem involves the sectional curvature.From the standard formula for the sectional curvature of Lie groups, see for instance [2] and [3], we notice that the sectional curvature of with respect to σ is well defined on "many planes" of the Lie algebra Lie( ).We also observe that the "finite dimensional formula" for the sectional curvature through the structure coefficients of Lie( ), [8,Lemma 1.1], converges on the previous planes to the same sectional curvature obtained by [2,Theorem 5].Broadly speaking, we may think that the convergence of the sectional curvature in Milnor's paper [8] could be interpreted as a computation of sectional curvature of through a finite dimensional approximation by an orthonormal basis.On the other side, we also observe that this convergence does not hold on all 2-dimensional subspaces of Lie( ), as shown in Remark 4.1.In addition, according to Proposition 4.2, we can also prove that our sectional curvature is discontinuous exactly at the plane where it cannot be defined.
Theorem 1.3.Let be the infinite dimensional Heisenberg group equipped with the left invariant weak Riemannian metric σ.Then there exists two sequences of orthonormal vectors a 1 j , a 2 j ∈ Lie( ) and b ∈ Lie( ) The numbers K σ (a 1 j , a 2 j ) and K σ (a 1 j , b) are the sectional curvatures of the planes of Lie( ) spanned by the orthonormal bases (a 1 j , a 2 j ) and (a 1 j , b).
The proof of this theorem is provided in Section 4, where also more information on the vectors a 1 j , a 2 j and b can be found.Inspecting the proofs of Theorem 3.2 and Theorem 1.3 another interesting phenomenon appears.The curves whose lengths converge to zero and that connect two distinct points are precisely contained in the span of the planes where the sectional curvature blows-up.

Preliminary notions
We denote by ℓ 2 the linear space of all real and square summable sequences.Its scalar product •, • has associated norm x = ∞ j=1 x 2 j for any element x = ∞ j=1 x j e j .The set of unit vectors {e j : j ≥ 1} denotes the canonical orthonormal basis of ℓ 2 .For each integer n ≥ 1, the element e n of ℓ 2 has n-th entry equal to 1 and all the others are zero.
We consider ℓ 2 × ℓ 2 × R endowed with its standard structure of product of Hilbert spaces.We also equip this space with a noncommutative Lie group operation We denote by the Hilbert Lie group arising from the previous group operation, that is the infinite dimensional Heisenberg group modelled on the Hilbert space ℓ 2 × ℓ 2 × R. The previous Lie group operation yields the Lie product )) (0, 0, 1), that makes also an infinite dimensional Heisenberg Lie algebra.For each p ∈ , we denote by L p : → the left multiplication by p, defined as L p (r) = p • r for all r ∈ .The group operation gives the following simple formula for the differential of L p at a point q, namely We have used a canonical identification between T q and , being a Hilbert manifold equipped with a structure of topological vector space.We also notice that our formula for the differential (dL p ) q does not depend on the point q.
2.1.Left invariant weak Riemannian metrics.We consider a continuous scalar product on the tangent space T 0 of at the origin.Then for every p ∈ and v, w ∈ T p the following scalar product (4) defines a left invariant weak Riemannian metric σ on .If for any piecewise smooth curve γ : [0, 1] → we define its Riemannian length as It is plain to check that d is left invariant, is symmetric and satisfies the triangle inequality.
Taking into account the canonical identification between and T 0 , the set ℓ 2 × ℓ 2 × {0} can be seen as a closed subspace of T 0 , that we denote by H 0 .Then we obtain a left invariant horizontal subbundle, denoted by H , whose fibers are and the previous condition corresponds to the equality We have a precise formula to define the horizontal curves associated to H .They are continuous and piecewise smooth curves γ : [0, 1] → of the form γ = (γ 1 , γ 2 , γ 3 ) ∈ , such that for almost every t ∈ [0, 1] we have The previous differential constraint means that γ(t) ∈ H γ(t) .
On the horizontal fibers H p of H we can fix a scalar product.A left invariant weak sub-Riemannian metric g on H is defined by a continuous inner product such that for all p ∈ and v, w ∈ H p we have (7) g p (v, w) = g 0 (dL p −1 ) p v, (dL p −1 ) p w = g 0 (dL −p ) p v, (dL −p ) p w .
The associated weak sub-Riemannian norm is denoted by • g and the length of a horizontal curve γ : [0, 1] → is defined by For any couple of points in , it is easy to construct a piecewise smooth horizontal curve that connects them, hence the following sub-Riemannian distance is finite for every couple of points p 1 , p 2 ∈ , hence we have ρ : × → [0, +∞).One may easily observe that ρ is left invariant, symmetric and satisfies the triangle inequality.

Degenerate geodesic distances in the infinite dimensional Heisenberg group
This section is devoted to the construction of special left invariant weak Riemannian (and sub-Riemannian) metrics that yield degenerate geodesic distances.
We introduce the linear and continuous operator A : ℓ 2 → ℓ 2 , which associates to each x ∈ ℓ 2 of components (x k ) k≥1 the element Ax ∈ ℓ 2 , whose k-th component is (Ax) k = x k /k.Then we define the scalar product η : for all v, w ∈ ℓ 2 .We use η to define the new scalar product (9) g for every (v 1 , v 2 ), (w 1 , w 2 ) ∈ ℓ 2 × ℓ 2 .By our identification, g 0 can be seen as a scalar product on H 0 , so that using (7) we obtain a left invariant weak sub-Riemannian metric g on .We follow the notation of the previous section, denoting by ρ the special sub-Riemannian distance associated to this choice of g through formula (8).
To obtain a left invariant weak Riemannian metric σ on , we extend g 0 as follows for every (v 1 , v 2 , v 3 ), (w 1 , w 2 , w 3 ) ∈ T 0 , where σ 0 : T 0 × T 0 → R. From (4), the scalar product in (10) immediately defines a left invariant weak Riemannian metric σ on .The Riemannian distance associated to σ through (5) will be denoted by d.From the definitions of g, d and ρ, one immediately observes that d ≤ ρ.
We consider the component γ i 0 = ∞ j=1 γ i 0 j e j and the following inequalities In particular, we have shown that The previous computation also shows that both d and ρ are actually distances, if restricted to any hyperplane ℓ 2 × ℓ 2 × {κ} with κ ∈ R.
We are now in a position to prove the following theorem.
Theorem 3.2.There exist a left invariant weak sub-Riemannian metric and a left invariant weak Riemannian metric on such that their associated geodesic distances are not positive on all couples of distinct points.
4. On the sectional curvature of a weak Riemannian Heisenberg group In this section, we study the sectional curvature of equipped with the weak Riemannian metric σ.From (10) we recall the formula For every positive integer j, we use the notation e 1 j = (e j , 0, 0), e 2 j = (0, e j , 0) and e 3 = (0, 0, 1), to indicate the standard orthonormal basis of seen as the Hilbert space ℓ 2 × ℓ 2 × R.
Since is connected, simply connected and nilpotent, by [9, Proposition IV.2.7], we can identify the vectors e i j and e 3 with the corresponding left invariant vector fields of Lie( ).Such identification is used to find the sectional curvature of , since it can be computed on planes of Lie( ).From (3), we have the formulas (19) [e 1 i , e 2 j ] = 2δ i j e 3 and [e l i , e l j ] = 0 for all i, j ≥ 1 and l = 1, 2. We consider a Lie algebra Lie(G) of a Fréchet Lie group G equipped with a weak Riemannian metric •, • .Following [2,Theorem 5], the point to compute the sectional curvature K(X, Y) of a plane in Lie(G) spanned by the orthonomal vectors X, Y in Lie(G) is to find the adjoint For a strong Riemannian metric, [1,Definition [5.2.12], the existence of B(X, Y) is always ensured, but not for any weak Riemannian metric.
From formula (53) of [2], we have where we define The proof of Theorem 1.3 follows from the application of (21) with respect to σ to suitable choices of planes.We denote by •, • σ the scalar product induced by the left invariant weak Riemannian metric σ on Lie( ).The associated norm on Lie( ) is denoted by • σ .We assume that for X, Y ∈ Lie( ) the adjoint with respect to σ exists.As a result, for Z ∈ Lie( ), by formula (3), we have where π : → ℓ 2 × ℓ 2 is the canonical projection defined by X = (π(X), x 3 ) = (π(X), 0) + x 3 e 3 .
We use the fixed orthonormal basis e 1 j , e 2 j , e 3 of with respect to the standard Hilbert product of Formula (24) yields (25) for arbitrary Z = Z 3 e 3 + ∞ j=1 Z 1 j e 1 j + Z 2 j e 2 j .In the case X = π(X), formula (25) shows the existence of ad(Y) * (π(X)) and yields Writing The assumption about the existence of B σ (e 3 , Y) corresponds to the convergence of its series.The next remark shows a choice of Y for which the series (29) does not converge.
Remark 4.1.We consider the vector for which the series (29) representing B σ (e 3 , W) does not converge.Clearly from (21) the sectional curvature K σ (e 3 , W) cannot be defined.
The previous remarks suggests that actually our sectional curvature is discontinuous.
Proposition 4.2.We consider the orthonormal elements W k , e 3 ∈ Lie( ) with k ≥ 1 and As the subspace span W k , e 3 converges to span{W ∞ , e 3 } for k → ∞, with The convergence of span W k , e 3 to span{W ∞ , e 3 } is considered in the Grassmannian of the 2-dimensional planes contained in Lie( ).
Finally, by formula (21), we have proved that as k → ∞.This concludes the proof.
Proof of Theorem 1.3.Following the notation of the present section, we define a 1 j = je 1 j and a 2 j = je 2 j of Lie( ), that are orthonormal with respect to •, • σ and do not commute.To apply (21) for finding K σ (a 1 j , a 2 j ), we use ( 22) and (23).Due to (26), we get B σ (a 1 j , a 2 j ) = B σ (a 2 j , a 1 j ) = 0.