Vanishing Geodesic Distance

on Spaces of Submanifolds and Diffeomorphisms

Peter W. Michor, David Mumford

Received: March 18, 2005

Communicated by Eckard Meinrenken

Abstract. The L^{2}-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L^{2}-metric.

2000 Mathematics Subject Classification: Primary 58B20, 58D15, 58E12

1. Introduction

In [10] we studied the L^{2}-Riemannian metric on the space of all immersions S^{1} → ℝ^{2}. This metric is invariant under the group Diff(S^{1}) and we found that it induces vanishing geodesic distance on the quotient space Imm(S^{1}, ℝ^{2})/ Diff(S^{1}). In this paper we extend this result to the general situation Imm(M, N)/ Diff(M) for any compact manifold M and Riemannian manifold (N, g) with dim N > dim M. On the open subset Emb(M, N)/ Diff(M), which may be identified with the space of all submanifolds of diffeomorphism type M in N (the non-linear Grassmanian or differentiable ‘Chow’ variety) this says that the infinite dimensional analog of the Fubini Study metric induces vanishing geodesic distance. The picture that emerges for these infinitedimensional manifolds is quite interesting: there are simple expressions for the Christoffel symbols and curvature tensor, the geodesic equations are simple and of hyperbolic type and, at least in the case of plane curves, the geodesic spray exists locally. But the curvature is positive and unbounded in some high frequency directions, so these spaces wrap up on themselves arbitrarily tightly, allowing the infimum of path lengths between two given points to be zero.

We also carry over to the general case the stronger metric from [10] which weights the L^{2} metric using the second fundamental form. It turns out that we have only to use the mean curvature in order to get positive geodesic distances, hence a good topological metric on the space Emb(S^{1}, ℝ^{2})/ Diff(S^{1}). The reason is that the first variation of the volume of a submanifold depends on the mean curvature and the key step is showing that the square root of the volume of M is Lipschitz in our stronger metric. The formula for this metric is:

G_{f}^{A}(h, k) := ∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2})g(h, k) vol(f^{∗}g)

where S^{f} is the second fundamental form of the immersion f; section 3 contains the relevant estimates. In section 4 we also compute the sectional curvature of the L^{2}-metric in the hope to relate the vanishing of the geodesic distance to unbounded positivity of the sectional curvature: by going through ever more positively curved parts of the space we can find ever shorter curves between any two submanifolds.

In the final section 5 we show that the vanishing of the geodesic distance also occurs on the Lie group of all diffeomorphisms on each connected Riemannian manifold. Short paths between any 2 diffeomorphisms are constructed by using rapidly moving compression waves in which individual points are trapped for relatively long times. We compute the sectional curvature also in this case.

2. The manifold of immersions

2.1. Conventions. Let M be a compact smooth connected manifold of dimension m ≥ 1 and let (N, g) be a connected Riemannian manifold of dimension n > m. We shall use the following spaces and manifolds of smooth mappings.

Diff(M), the regular Lie group ([8], 38.4) of all diffeomorphisms of M.

Diff_{x_{0}}(M), the subgroup of diffeomorphisms fixing x_{0} ∈ M.

Emb = Emb(M, N), the manifold of all smooth embeddings M → N.

Imm = Imm(M, N), the manifold of all smooth immersions M → N. For an immersion f the tangent space with foot point f is given by T_{f} Imm(M, N) = C_{f}^{∞}(M, T N) = Γ(f^{∗}T N), the space of all vector fields along f.

Imm_{f} = Imm_{f}(M, N), the manifold of all smooth free immersions M → N, i.e., those with trivial isotropy group for the right action of Diff(M) on Imm(M, N).

B_{e} = B_{e}(M, N) = Emb(M, N)/ Diff(M), the manifold of submanifolds of type M in N, the base of a smooth principal bundle, see 2.2.

B_{i} = B_{i}(M, N) = Imm(M, N)/ Diff(M), an infinite dimensional ‘orbifold’, whose points are, roughly speaking, smooth immersed submanifolds of type M in N, see 2.4.

B_{i,f} = B_{i,f}(M, N) = Imm_{f}(M, ℝ^{2})/ Diff(M), a manifold, the base of a principal fiber bundle, see 2.3.

For a smooth curve f : ℝ → C^{∞}(M, N) corresponding to a mapping f : ℝ × M → N, we shall denote by T f the curve of tangent mappings, so that T f(t)(X_{x}) = T_{x}(f(t, )).X_{x}. The time derivative will be denoted by either ∂_{t}f = f_{t} : ℝ × M → T N.

2.2. The principal bundle of embeddings Emb(M, N). We recall some basic results whose proof can be found in [8]):

(A) The set Emb(M, N) of all smooth embeddings M → N is an open subset of the smooth Fréchet manifold C^{∞}(M, N) of all smooth mappings M → N with the C^{∞}-topology. It is the total space of a smooth principal bundle π : Emb(M, N) → B_{e}(M, N) with structure group Diff(M), the smooth regular Lie group group of all diffeomorphisms of M, whose base B_{e}(M, N) is the smooth Fréchet manifold of all submanifolds of N of type M, i.e., the smooth manifold of all simple closed curves in N. ([8], 44.1)

(B) This principal bundle admits a smooth principal connection described by the horizontal bundle whose fiber N_{c} over c consists of all vector fields h along f such that g(h, T f) = 0. The parallel transport for this connection exists and is smooth. ([8], 39.1 and 43.1)

2.3. Free immersions. The manifold Imm(M, N) of all immersions M → N is an open set in the manifold C^{∞}(M, N) and thus itself a smooth manifold. An immersion f : M → N is called free if Diff(M) acts freely on it, i.e., f ∘ φ = c for φ ∈ Diff(M) implies φ = Id. We have the following results:

• If φ ∈ Diff(M) has a fixed point and if f ∘ φ = f for some immersion f then φ = Id. This is ([4], 1.3).

• If for f ∈ Imm(M, N) there is a point x ∈ f(M) with only one preimage then f is a free immersion. This is ([4], 1.4). There exist free immersions without such points.

• The manifold B_{i,f}(M, N) ([4], 1.5) The set Imm_{f}(M, N) of all free immersions is open in C^{∞}(M, N) and thus a smooth submanifold. The projection

π : Imm_{f}(M, N) → \frac{Imm_{f}(M, N)}{Diff(M)} =: B_{i,f}(M, N)

onto a Hausdorff smooth manifold is a smooth principal fibration with structure group Diff(M). By ([8], 39.1 and 43.1) this fibration admits a smooth principal connection described by the horizontal bundle with fiber N_{c} consisting of all vector fields h along f such that g(h, T f) = 0. This connection admits a smooth parallel transport over each smooth curve in the base manifold.

We might view Imm_{f}(M, N) as the nonlinear Stiefel manifold of parametrized submanifolds of type M in N and consequently B_{i,f}(M, N) as the nonlinear Grassmannian of unparametrized submanifolds of type M.

2.4. Non free immersions. Any immersion is proper since M is compact and thus by ([4], 2.1) the orbit space B_{i}(M, N) = Imm(M, N)/ Diff(M) is Hausdorff. Moreover, by ([4], 3.1 and 3.2) for any immersion f the isotropy group Diff(M)_{f} is a finite group which acts as group of covering transformations for a finite covering q_{c} : M → M̄ such that f factors over q_{c} to a free immersion f̄: M̄ → N with f̄ ∘ q_{c} = f. Thus the subgroup Diff_{x_{0}}(M) of all diffeomorphisms φ fixing x_{0} ∈ M acts freely on Imm(M, N). Moreover, for each f ∈ Imm the submanifold Q(f) from 4.4, (1) is a slice in a strong sense:

• Q(f) is invariant under the isotropy group Diff(M)_{f}.

• If Q(f)∘φ∩Q(f) ≠ ∅ for φ ∈ Diff(M) then φ is already in the isotropy group φ ∈ Diff(M)_{f}.

• Q(f) ∘ Diff(M) is an invariant open neigbourhood of the orbit f ∘ Diff(M) in Imm(M, N) which admits a smooth retraction r onto the orbit. The fiber r^{−1}(f ∘ φ) equals Q(f ∘ φ).

Note that also the action

Imm(M, N) × Diff(M) → Imm(M, N) × Imm(M, N), (f, φ) ↦ (f, f ∘ φ)

is proper so that all assumptions and conclusions of Palais’ slice theorem [13] hold. This results show that the orbit space B_{i}(M, N) has only singularities of orbifold type times a Fréchet space. We may call the space B_{i}(M, N) an infinite dimensional orbifold. The projection π : Imm(M, N) → B_{i}(M, N) = Imm(M, N)/ Diff(M) is a submersion off the singular points and has only mild singularities at the singular strata. The normal bundle N_{f} mentioned in 2.2 is well defined and is a smooth vector subbundle of the tangent bundle. We do not have a principal bundle and thus no principal connections, but we can prove the main consequence, the existence of horizontal paths, directly:

2.5. Proposition. For any smooth path f in Imm(M, N) there exists a smooth path φ in Diff(M) with φ(t, ) = Id_{M} depending smoothly on f such that the path h given by h(t, θ) = f(t, φ(t, θ)) is horizontal: g(h_{t}, T h) = 0.

Proof. Let us write h = f ∘ φ for h(t, x) = f(t, φ(t, x)), etc. We look for φ as the integral curve of a time dependent vector field ξ(t, x) on M, given by ∂_{t}φ = ξ ∘ φ. We want the following expression to vanish:

g(∂_{t}(f ∘ φ), T (f ∘ φ)) = g((∂_{t}f ∘ φ + (T f ∘ φ).∂_{t}φ, (T f ∘ φ).T φ)

= (g(∂_{t}f, T f) ∘ φ).T φ + g((T f ∘ φ)(ξ ∘ φ), (T f ∘ φ).T φ)

= ((g(∂_{t}f, T f) + g(T f.ξ, T f)) ∘ φ).T φ

Since T φ is everywhere invertible we get

0 = g(∂_{t}(f ∘ φ), T (f ∘ φ)) ⟺ 0 = g(∂_{t}f, T f) + g(T f.ξ, T f)

and the latter equation determines the non-autonomous vector field ξ uniquely. □

2.6. Curvatures of an immersion. Consider a fixed immersion f ∈ Imm(M, N). The normal bundle N(f) = T f^{⊥} ⊂ f^{∗}T N → M has fibers N(f)_{x} = {Y ∈ T_{f(x)}N : g(Y, T_{x}f.X) = 0 for all X ∈ T_{x}M}. Every vector field h : M → T N along f then splits as h = T f.h^{⊤} + h^{⊥} into its tangential component h^{⊤} ∈ 𝔛(M) and its normal component h^{⊥} ∈ Γ(N(f)).

Let ∇^{g} be the Levi-Civita covariant derivative of g on N and let ∇^{f^{∗}g} the Levi-Civita covariant derivative of the pullback metric f^{∗}g on M. The shape operator or second fundamental form S^{f} ∈ Γ(S^{2}T^{∗}M ⊗N(f)) of f is then given by

(1)  S^{f}(X, Y) = ∇_{X}^{g}(T f.Y) − T f.∇_{X}^{f^{∗}g}Y for X, Y ∈ 𝔛(M).

It splits into the following irreducible components under the action of the group O(T_{x}M) × O(N(f)_{x}): the mean curvature Tr^{f^{∗}g}(S^{f}) = Tr((f^{∗}g)^{−1} ∘ S^{f}) ∈ Γ(N(f)) and the trace free shape operator S_{0}^{f} = S^{f} −Tr^{f^{∗}g}(S^{f}). For X ∈ 𝔛(M) and ξ ∈ Γ(N(f)), i.e., a normal vector field along f, we may also split ∇_{X}^{g}ξ into the components which are tangential and normal to T f.T M,

(2)  ∇_{X}^{g}ξ = −T f.L_{ξ}^{f}(X) + ∇_{X}^{N(f)}ξ

where ∇^{N(f)} is the induced connection in the normal bundle respecting the metric g^{N(f)} induced by g, and where the Weingarten tensor field L^{f} ∈ Γ(N(f)^{∗} ⊗ T^{∗}M ⊗ T M) corresponds to the shape operator via the formula

(3)  (f^{∗}g)(L_{ξ}^{f}(X), Y) = g^{N(f)}(S^{f}(X, Y), ξ).

Let us also split the Riemann curvature R^{g} into tangential and normal parts: For X_{i} ∈ 𝔛(M) or T_{x}M we have (theorema egregium):

g(R^{g}(T f.X_{1}, T f.X_{2})(T f.X_{3}), T f.X_{4}) = (f^{∗}g)(R^{f^{∗}g}(X_{1}, X_{2})X_{3}, X_{4})+

+ g^{N(f)}(S^{f}(X_{1}, X_{3}), S^{f}(X_{2}, X_{4})) − g^{N(f)}(S^{f}(X_{2}, X_{3}), S^{f}(X_{1}, X_{4})). (4)

The normal part of R^{g} is then given by (Codazzi-Mainardi equation):

(R^{g}(T f.X_{1}, T f.X_{2})(T f.X_{3}))^{⊥} =

= (∇_{X_{1}}^{N(f)⊗T^{∗}M⊗T^{∗}M}S^{f})(X_{2}, X_{3}) − (∇_{X_{2}}^{N(f)⊗T^{∗}M⊗T^{∗}M}S^{f})(X_{1}, X_{3}). (5)

2.7. Volumes of an immersion. For an immersion f ∈ Imm(M, N), we consider the volume density vol^{g}(f) = vol(f^{∗}g) ∈ Vol(M) on M given by the local formula vol^{g}(f)|_{U} = \sqrt{det((f^{∗}g)_{ij})}|du^{1} ∧ ⋯ ∧ du^{m}| for any chart (U, u : U → ℝ^{m}) of M, and the induced volume function Vol^{g} : Imm(M, N) → ℝ_{>0} which is given by Vol^{g}(f) = ∫_{M} vol(f^{∗}g). The tangent mapping of vol : Γ(S_{>0}^{2}T^{∗}M) → Vol(M) is given by d vol(γ)(η) = \frac{1}{2} Tr(γ^{−1}.η) vol(γ). We consider the pullback mapping P_{g} : f ↦ f^{∗}g, P_{g} : Imm(M, N) → Γ(S_{>0}^{2}T^{∗}M). A version of the following lemma is [7], 1.6.

Lemma. The derivative of vol^{g} = vol ∘P_{g} : Imm(M, N) → Vol(M) is

d vol^{g}(h) = d(vol ∘P_{g})(h)
= − Tr^{f^{∗}g}(g(S^{f}, h^{⊥})) vol(f^{∗}g) + \frac{1}{2} Tr^{f^{∗}g}(L_{h^{⊤}}(f^{∗}g)) vol(f^{∗}g).
= −(g(Tr^{f^{∗}g}(S^{f}), h^{⊥})) vol(f^{∗}g) + div^{f^{g}}(h^{⊤}) vol(f^{∗}g).

Proof. We consider a curve t ↦ f(t, ) in Imm with ∂_{t}|_{0}f = h. We also use a chart (U, u : U → ℝ^{m}) on M. Then we have

f^{∗}g|_{U} = ∑_{i,j}(f^{∗}g)_{ij}du^{i} ⊗ du^{j} = ∑_{i,j} g(T f.∂_{u_{i}}, T f.∂_{u^{j}})du^{i} ⊗ du^{j}

∂_{t} vol^{g}(f)|_{U} = \frac{det((f^{∗}g)_{ij})(f^{∗}g)^{kl}∂_{t}(f^{∗}g)_{lk}}{2\sqrt{det((f^{∗}g)_{ij})}} |du^{1} ∧ ⋯ ∧ du^{m}|

where

∂_{t}(f^{∗}g)_{ij} = ∂_{t}g(T f.∂_{u^{i}}, T f.∂_{u^{j}})
= g(∇_{∂_{t}}^{g}(T f.∂_{u^{i}}), T f.∂_{u^{j}}) + g(∂_{u^{i}}, ∇_{∂_{t}}^{g}(T f.∂_{u^{j}}))
g(∇_{∂_{t}}^{g}T f.∂_{u^{i}}, T f.∂_{u^{j}}) = g(∇_{∂_{u^{i}}}^{g} T f.∂_{t} + T f. Tor +T f.[∂_{t}, ∂_{u^{i}}], T f.∂_{u^{j}})
= g(∇_{∂_{u^{i}}}^{g} (T f.∂_{t})^{⊥}, T f.∂_{u^{j}}) + g(∇_{∂_{u^{i}}}^{g} (T f.∂_{t})^{⊤}, T f.∂_{u^{j}})
g(∇_{∂_{u^{i}}}^{g} (∂_{t}f)^{⊥}, T f.∂_{u^{j}}) = g(−T f.L_{(∂_{t}f)^{⊥}}∂_{u^{i}}, T f.∂_{u^{j}})+g(∇_{∂_{u^{i}}}^{N(f)}(∂_{t}f)^{⊥}, T f.∂_{u^{j}})
= −(f^{∗}g)(L_{(∂_{t}f)^{⊥}}∂_{u^{i}}, ∂_{u^{j}})
= −g(S^{f}(∂_{u^{i}}, ∂_{u^{j}}), (∂_{t}f)^{⊥})
g(∇_{∂_{u^{i}}}^{g} (∂_{t}f)^{⊤}, T f.∂_{u^{j}}) = (f^{∗}g)(∇_{∂_{u^{i}}}^{f^{∗}g}(∂_{t}f)^{⊤}, ∂_{u^{j}}) + 0
= (f^{∗}g)(∇_{(∂_{t}f)^{⊤}}^{f^{∗}g}∂_{u^{i}} + Tor −[(∂_{t}f)^{⊤}, ∂_{u^{i}}], ∂_{u^{j}}),
∂_{t}(f^{∗}g)_{ij} = −2g(S^{f}(∂_{u^{i}}, ∂_{u^{j}}), (∂_{t}f)^{⊥}) + (L_{(∂_{t}f)^{⊤}}(f^{∗}g))(∂_{u^{i}}, ∂_{u^{j}})

This proves the first formula. For the second one note that

\frac{1}{2} Tr((f^{∗}g)^{−1}L_{h^{⊤}}(f^{∗}g)) vol(f^{∗}g) = L_{h^{⊤}}(vol(f^{∗}g)) = div^{f^{∗}g}(h^{⊤}) vol(f^{∗}g). □

3. Metrics on spaces of mappings

3.1. The metric G^{A}. Let h, k ∈ C_{f}^{∞}(M, T N) be two tangent vectors with foot point f ∈ Imm(M, N), i.e., vector fields along f. Let the induced volume density be vol(f^{∗}g). We consider the following weak Riemannian metric on Imm(M, N), for a constant A ≥ 0:

G_{f}^{A}(h, k) := ∫_{M} (1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2}) g(h, k) vol(f^{∗}g)

where Tr^{f^{∗}g}(S^{f}) ∈ N(f) is the mean curvature, a section of the normal bundle, and ∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}} is its norm. The metric G^{A} is invariant for the action of Diff(M). This makes the map π : Imm(M, N) → B_{i}(M, N) into a Riemannian submersion (off the singularities of B_{i}(M, N)).

Now we can determine the bundle N → Imm(M, N) of tangent vectors which are normal to the Diff(M)-orbits. The tangent vectors to the orbits are T_{f}(f ∘ Diff(M)) = {T f.ξ : ξ ∈ 𝔛(M)}. Inserting this for k into the expression of the metric G we see that

N_{f} = {h ∈ C^{∞}(M, T N) : g(h, T f) = 0}
= Γ(N(f)),

the space of sections of the normal bundle. This is independent of A.

A tangent vector h ∈ T_{f} Imm(M, N) = C_{f}^{∞}(M, T N) = Γ(f^{∗}T N) has an orthonormal decomposition

h = h^{⊤} + h^{⊥} ∈ T_{f}(f ∘ Diff^{+}(M)) ⊕ N_{f}

into smooth tangential and normal components.

Since the Riemannian metric G^{A} on Imm(M, N) is invariant under the action of Diff(M) it induces a metric on the quotient B_{i}(M, N) as follows. For any F_{0}, F_{1} ∈ B_{i}, consider all liftings f_{0}, f_{1} ∈ Imm such that π(f_{0}) = F_{0}, π(f_{1}) = F_{1} and all smooth curves t ↦ f(t, ) in Imm(M, N) with f(0, ·) = f_{0} and f(1, ·) = f_{1}. Since the metric G^{A} is invariant under the action of Diff(M) the arc-length of the curve t ↦ π(f(t, ·)) in B_{i}(M, N) is given by

L_{G^{A}}^{hor}(f) := L_{G^{A}}(π(f(t, ·))) =
= ∫_{0}^{1} \sqrt{G_{π(f)}^{A}(T_{f}π.f_{t}, T_{f}π.f_{t})} dt = ∫_{0}^{1} \sqrt{G_{f}^{A}(f_{t}^{⊥}, f_{t}^{⊥})} dt =

= ∫_{0}^{1}(∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2})g(f_{t}^{⊥}, f_{t}^{⊥}) vol(f^{∗}g))^{\frac{1}{2}} dt

In fact the last computation only makes sense on B_{i,f}(M, N) but we take it as a motivation. The metric on B_{i}(M, N) is defined by taking the infimum of this over all paths f (and all lifts f_{0}, f_{1}):

dist_{G^{A}}^{B_{i}}(F_{1}, F_{2}) = inf_{f} L_{G^{A}}^{hor}(f).

3.2. Theorem. Let A = 0. For f_{0}, f_{1} ∈ Imm(M, N) there exists always a path t ↦ f(t, ·) in Imm(M, N) with f(0, ·) = f_{0} and π(f(1, ·)) = π(f_{1}) such that L_{G^{0}}^{hor}(f) is arbitrarily small.

Proof. Take a path f(t, θ) in Imm(M, N) from f_{0} to f_{1} and make it horizontal using 2.4 so that that g(f_{t}, T f) = 0; this forces a reparametrization on f_{1}.

Let α : M → [0, 1] be a surjective Morse function whose singular values are all contained in the set \{\frac{k}{2N} : 0 ≤ k ≤ 2N\} for some integer N. We shall use integers n below and we shall use only multiples of N.

Then the level sets M_{r} := {x ∈ M : α(x) = r} are of Lebesque measure 0. We shall also need the slices M_{r_{1},r_{2}} := {x ∈ M : r_{1} ≤ α(x) ≤ r_{2}}. Since M is compact there exists a constant C such that the following estimate holds uniformly in t:

∫_{M_{r_{1},r_{2}}} vol(f(t, )^{∗}g) ≤ C(r_{2} − r_{1}) ∫_{M} vol(f(t, )^{∗}g)

Let f̃(t, x) = f(φ(t, α(x)), x) where φ : [0, 1] × [0, 1] → [0, 1] is given as in [10], 3.10 (which also contains a figure illustrating the construction) by

φ(t, α)=\begin{cases}2t(2nα − 2k) & for 0 ≤ t ≤ 1/2, \frac{2k}{2n} ≤ α ≤ \frac{2k+1}{2n} \\ 2t(2k + 2 − 2nα) & for 0 ≤ t ≤ 1/2, \frac{2k+1}{2n} ≤ α ≤ \frac{2k+2}{2n} \\ 2t − 1 + 2(1 − t)(2nα − 2k) & for 1/2 ≤ t ≤ 1, \frac{2k}{2n} ≤ α ≤ \frac{2k+1}{2n} \\ 2t − 1 + 2(1 − t)(2k + 2 − 2nα) & for 1/2 ≤ t ≤ 1, \frac{2k+1}{2n} ≤ α ≤ \frac{2k+2}{2n}. \end{cases}

Then we get T f̃ = φ_{α}.dα.f_{t} + T f and f̃_{t} = φ_{t}.f_{t} where

φ_{α} = \begin{cases}+4nt \\ −4nt \\ +4n(1 − t) \\ −4n(1 − t) \end{cases} , φ_{t} = \begin{cases}4nα − 4k \\ 4k + 4 − 4nα \\ 2 − 4nα + 4k \\ −(2 − 4nα + 4k) \end{cases} .

We use horizontality g(f_{t}, T f) = 0 to determine f̃_{t}^{⊥} = f̃_{t} + T f̃(X) where X ∈ T M satisfies 0 = g(f̃_{t} + T f̃(X), T f̃(ξ)) for all ξ ∈ T M. We also use

dα(ξ) = f^{∗}g(grad^{f^{∗}g} α, ξ) = g(T f(grad^{f^{∗}g} α), T f(ξ))

and get

0 = g(f̃_{t} + T f̃(X), T f̃(ξ))

= g(φ_{t}f_{t} + φ_{α}dα(X)f_{t} + T f(X), φ_{α}dα(ξ)f_{t} + T f(ξ))

= φ_{t}.φ_{α}.(f^{∗}g)(grad^{f^{∗}g} α, ξ)∥f_{t}∥_{g}^{2}+

+ φ_{α}^{2}.(f^{∗}g)(grad^{f^{∗}g} α, X).(f^{∗}g)(grad^{f^{∗}g} α, ξ)∥f_{t}∥_{g}^{2} + g(T f(X), T f(ξ))

= (φ_{t}.φ_{α} + φ_{α}^{2}.(f^{∗}g)(grad^{f^{∗}g} α, X))∥f_{t}∥_{g}^{2}(f^{∗}g)(grad^{f^{∗}g} α, ξ) + (f^{∗}g)(X, ξ)

This implies that X = λ grad^{f^{∗}g} α for a function λ and in fact we get

f̃_{t}^{⊥} = \frac{φ_{t}}{1 + φ_{α}^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}} f_{t} − \frac{φ_{t}φ_{α}∥f_{t}∥_{g}^{2}}{1 + φ_{α}^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}} T f(grad^{f^{∗}g} α)

and

∥f̃_{t}∥_{g}^{2} = \frac{φ_{t}^{2}∥f_{t}∥_{g}^{2}}{1 + φ_{α}^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}

From T f̃ = φ_{α}.dα.f_{t} + T f and g(f_{t}, T f) = 0 we get for the volume form

vol(f̃^{∗}g) = \sqrt{1 + φ_{α}^{2} ∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}} vol(f^{∗}g).

For the horizontal length we get

L^{hor}(f̃) = ∫_{0}^{1}(∫_{M} ∥f̃_{t}^{⊥}∥_{g}^{2} vol(f̃^{∗}g))^{\frac{1}{2}} dt =

= ∫_{0}^{1}(∫_{M} \frac{φ_{t}^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + φ_{α}^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g))^{\frac{1}{2}} dt =

= ∫0^{\frac{1}{2}} (∑_{k=0}^{n−1}(∫_{M_{\frac{2k}{2n} , \frac{2k+1}{2n}}} \frac{(4nα − 4k)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4nt)^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g)+

+ ∫_{M_{\frac{2k+1}{2n} , \frac{2k+2}{2n}}} \frac{(4k + 4 − 4nα)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4nt)^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g)))^{\frac{1}{2}} dt+

+ ∫ 1_{\frac{1}{2}}(∑_{k=0}^{n−1}(∫_{M_{\frac{2k}{2n} , \frac{2k+1}{2n}}} \frac{(2 − 4nα + 4k)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4n(1 − t))^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g)+

+ ∫_{M_{\frac{2k+1}{2n} , \frac{2k+2}{2n}}} \frac{(2 − 4nα + 4k)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4n(1 − t))^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g)))^{\frac{1}{2}} dt

Let ε > 0. The function (t, x) ↦ ∥f_{t}(φ(t, α(x)), x)∥_{g}^{2} is uniformly bounded. On M_{\frac{2k}{2n} , \frac{2k+1}{2n}} the function 4nα − 4k has values in [0, 2]. Choose disjoint geodesic

balls centered at the finitely many singular values of the Morse function α of total f^{∗}g-volume < ε. Restricted to the union M_{sing} of these balls the integral above is O(1)ε. So we have to estimate the integrals on the complement M̃ = M \ M_{sing} where the function ∥dα∥_{f^{∗}g} is uniformly bounded from below by η > 0.

Let us estimate one of the sums above. We use the fact that the singular points of the Morse function α lie all on the boundaries of the sets M̃_{\frac{2k}{2n} , \frac{2k+1}{2n}} so that

we can transform the integrals as follows:

∑_{k=0}^{n−1} ∫M̃_{\frac{2k}{2n} , \frac{2k+1}{2n}} \frac{(4nα − 4k)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4nt)^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} vol(f^{∗}g) =

= ∑_{k=0}^{n−1} ∫ \frac{2k+1}{2n} ; \frac{2k}{2n} ∫M̃_{r} \frac{(4nr − 4k)^{2}∥f_{t}∥_{g}^{2}}{\sqrt{1 + (4nt)^{2}∥dα∥_{f^{∗}g}^{2}∥f_{t}∥_{g}^{2}}} \frac{vol(i_{r}^{∗}f^{∗}g)}{∥dα∥_{f^{∗}g}} dr

We estimate this sum of integrals: Consider first the set of all (t, r, x ∈ M_{r}) such that |f_{t}(φ(t, r), x)| < ε. There we estimate by

O(1).n.16n^{2}.ε^{2}.(r^{3}/3)|_{r=0}^{r=1/2n} = O(ε).

On the complementary set where |f_{t}(φ(t, r), x)| ≥ ε we estimate by

O(1).n.16n^{2}.\frac{1}{4ntη^{2}ε}(r^{3}/3)|_{r=0}^{r=1/2n} = O(\frac{1}{ntη^{2}ε})

which goes to 0 if n is large enough. The other sums of integrals can be estimated similarly, thus L^{hor}(f̃) goes to 0 for n → ∞. It is clear that one can approximate φ by a smooth function whithout changing the estimates essentially. □

3.3. A Lipschitz bound for the volume in G^{A}. We apply the Cauchy-Schwarz inequality to the derivative 2.7 of the volume Vol^{g}(f) along a curve t ↦ f(t, ) ∈ Imm(M, N):

∂_{t} Vol^{g}(f) = ∂_{t} ∫_{M} vol^{g}(f(t, )) = ∫_{M} d vol^{g}(f)(∂_{t}f)

= − ∫_{M} Tr^{f^{∗}g}(g(S^{f}, f_{t}^{⊥})) vol(f^{∗}g) ≤ |∫_{M} Tr^{f^{∗}g}(g(S^{f}, f_{t}^{⊥})) vol(f^{∗}g)|

≤ (∫_{M} 1^{2} vol(f^{∗}g))^{\frac{1}{2}} (∫_{M} Tr^{f^{∗}g}(g(S^{f}, f_{t}^{⊥}))^{2} vol(f^{∗}g))^{\frac{1}{2}}

≤ Vol^{g}(f)^{\frac{1}{2}} \frac{1}{\sqrt{A}}(∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2}) g(f_{t}^{⊥}, f_{t}^{⊥}) vol(f^{∗}g))^{\frac{1}{2}}

Thus

∂_{t}(\sqrt{Vol^{g}(f)}) = \frac{∂_{t} Vol^{g}(f)}{2\sqrt{Vol^{g}(f)}} ≤

≤ \frac{1}{2\sqrt{A}}(∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2}) g(f_{t}^{⊥}, f_{t}^{⊥}) vol(f^{∗}g))^{\frac{1}{2}}

and by using (3.1) we get

\sqrt{Vol^{g}(f_{1})} − \sqrt{Vol^{g}(f_{0})} = ∫_{0}^{1} ∂_{t}(\sqrt{Vol^{g}(f)}) dt

≤ \frac{1}{2\sqrt{A}} ∫_{0}^{1}(∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2}) g(f_{t}^{⊥}, f_{t}^{⊥}) vol(f^{∗}g))^{\frac{1}{2}} dt

= \frac{1}{2\sqrt{A}}L_{G^{A}}^{hor}(f).

If we take the infimum over all curves connecting f_{0} with the Diff(M)-orbit through f_{1} we get:

Proposition. Lipschitz continuity of \sqrt{Vol^{g}} : B_{i}(M, N) → ℝ_{≥0}. For F_{0} and F_{1} in B_{i}(M, N) = Imm(M, N)/ Diff(M) we have for A > 0:

\sqrt{Vol^{g}(F_{1})} − \sqrt{Vol^{g}(F_{0})} ≤ \frac{1}{2\sqrt{A}} dist_{G^{A}}^{B_{i}}(F_{1}, F_{2}).

3.4. Bounding the area swept by a path in B_{i}. We want to bound the area swept out by a curve starting from F_{0} to any immersed submanifold F_{1} nearby in our metric. First we use the Cauchy-Schwarz inequality in the Hilbert space L^{2}(M, vol(f(t, )^{∗}g)) to get

∫_{M} 1.∥f_{t}∥_{g} vol(f^{∗}g) = ⟨1, ∥f_{t}∥_{g}⟩_{L^{2}} ≤

≤ ∥1∥_{L^{2}}∥c_{t}∥_{L^{2}} = (∫_{M} vol(f^{∗}g))^{\frac{1}{2}} (∫_{M} |f_{t}| vol(f^{∗}g))^{\frac{1}{2}} .

Now we assume that the variation f(t, x) is horizontal, so that g(f_{t}, T f) = 0. Then L_{G^{A}}(f) = L_{G^{A}}^{hor}(f). We use this inequality and then the intermediate value theorem of integral calculus to obtain

L_{G^{A}}^{hor}(f) = L_{G^{A}}(f) = ∫_{0}^{1} \sqrt{G_{f}^{A}(f_{t}, f_{t})} dt

= ∫_{0}^{1}(∫_{M}(1 + A∥ Tr^{f^{∗}}(S^{f})∥_{f^{∗}g}^{2})∥f_{t}∥^{2} vol(f^{∗}g))^{\frac{1}{2}} dt

≥ ∫_{0}^{1}(∫_{M} ∥f_{t}∥^{2} vol(f^{∗}g))^{\frac{1}{2}} dt

≥ ∫_{0}^{1}(∫_{M} vol(f(t, )^{∗}g))^{− \frac{1}{2}} ∫_{M} ∥f_{t}(t, )∥_{g} vol(f(t, )^{∗}g) dt

= (∫_{M} vol(f(t_{0}, )^{∗}g))^{− \frac{1}{2}} ∫_{0}^{1} ∫_{M} ∥f_{t}(t, )∥_{g} vol(f(t, )^{∗}g) dt

for some intermediate value 0 ≤ t_{0} ≤ 1,

≥ \frac{1}{\sqrt{Vol^{g}(f(t_{0}, ))}} ∫_{[0,1]×M} vol^{m+1}(f^{∗}g)

Documenta Mathematica 10 (2005) 217–245

Proposition. Area swept out bound. If f is any path from F_{0} to F_{1}, then

(\begin{matrix}(m + 1) − volume of the region swept \\ out by the variation f \end{matrix}) ≤ max_{t} \sqrt{Vol^{g}(f(t, ))} · L_{G^{A}}^{hor}(f).

Together with the Lipschitz continuity 3.3 this shows that the geodesic distance inf L_{G^{A}}^{B_{i}} separates points, at least in the base space B(M, N) of embeddings.

3.5. Horizontal energy of a path as anisotropic volume. We consider a path t ↦ f(t, ) in Imm(M, N). It projects to a path π ∘ f in B_{i} whose energy is:

E_{G^{A}}(π ∘ f) = \frac{1}{2} ∫_{a}^{b} G_{π(f)}^{A}(T π.f_{t}, T π.f_{t}) dt = \frac{1}{2} ∫_{a}^{b} G_{f}^{A}(f_{t}^{⊥}, f_{t}^{⊥}) dt =

= \frac{1}{2} ∫_{a}^{b} ∫_{M}(1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2})g(f_{t}^{⊥}, f_{t}^{⊥}) vol(f^{∗}g) dt.

We now consider the graph γ_{f} : [a, b] × M ∋ (t, x) ↦ (t, f(t, x)) ∈ [a, b] × N of the path f and its image Γ_{f}, an immersed submanifold with boundary of ℝ × N. We want to describe the horizontal energy as a functional on the space of immersed submanifolds with fixed boundary, remembering the fibration of pr_{1} : ℝ × N → ℝ. We get:

E_{G^{A}}(π ∘ f) =

= \frac{1}{2} ∫_{[a,b]×M} (1 + A∥ Tr^{f^{∗}g}(S^{f})∥_{g^{N(f)}}^{2}) \frac{∥f_{t}^{⊥}∥^{2}}{\sqrt{1 + ∥f_{t}^{⊥}∥_{g}^{2}}} vol(γ_{f}^{∗}(dt^{2} + g))

Now ∥f_{t}^{⊥}∥_{g} depends only on the graph Γ_{f} and on the fibration over time, since any reparameterization of Γ_{f} which respects the fibration over time is of the form (t, x) ↦ (t, f(t, φ(t, x))) for some path φ in Diff(M) starting at the identity, and (∂_{t}|_{0}f(t, φ(t, x)))^{⊥} = f_{t}^{⊥}. So the above expression is intrinsic for the graph Γ_{f} and the fibration. In order to find a geodesic from the shape π(f(a, )) to the shape π(f(b, )) one has to find an immersed surface which is a critical point for the functional E_{G^{A}} above. This is a Plateau-problem with anisotropic volume.

4. The geodesic equation and the curvature on B_{i}

4.1. The geodesic equation of G^{0} in Imm(M, N). The energy of a curve t ↦ f(t, ) in Imm(M, N) for G^{0} is

E_{G^{0}}(f) = \frac{1}{2} ∫_{a}^{b} ∫_{M} g(f_{t}, f_{t}) vol(f^{∗}g).

Vanishing Geodesic Distance … 229

The geodesic equation for G^{0} (1)

∇_{∂_{t}}^{g}f_{t} + div^{f^{∗}g}(f_{t}^{⊤})f_{t} − g(f_{t}^{⊥}, Tr^{f^{∗}g}(S^{f}))f_{t}+
+ \frac{1}{2}T f. grad^{f^{∗}g}(∥f_{t}∥_{g}^{2}) + \frac{1}{2}∥f_{t}∥_{g}^{2} Tr^{f^{∗}g}(S^{f}) = 0

Proof. A different proof is in [7], 2.2. For a function a on M we shall use

∫_{M} a div^{f^{∗}g}(X) vol(f^{∗}g) = ∫_{M} aL_{X}(vol(f^{∗}g))

= ∫_{M} L_{X}(a vol(f^{∗}g)) − ∫_{M} L_{X}(a) vol(f^{∗}g) = − ∫_{M}(f^{∗}g)(grad^{f^{∗}g}(a), X) vol(f^{∗}g)

in calculating the first variation of the energy with fixed ends:

∂_{s}E_{G^{0}}(f) = \frac{1}{2} ∫_{a}^{b} ∫_{M}(∂_{s}g(f_{t}, f_{t}) vol(f^{∗}g) + g(f_{t}, f_{t}) ∂_{s} vol(f^{∗}g))dt

= ∫_{a}^{b} ∫_{M}(g(∇_{∂_{s}}^{g}f_{t}, f_{t}) vol(f^{∗}g) + \frac{1}{2}∥f_{t}∥_{g}^{2} div^{f^{∗}g}(f_{s}^{⊤}) vol(f^{∗}g)

− \frac{1}{2}∥f_{t}∥_{g}^{2}g(f_{s}^{⊥}, Tr^{f^{∗}g}(S^{f})) vol(f^{∗}g))dt

For the first summand we have:

∫_{a}^{b} ∫_{M} g(∇_{∂_{s}}^{g}f_{t}, f_{t}) vol(f^{∗}g) dt = ∫_{a}^{b} ∫_{M} g(∇_{∂_{t}}^{g}f_{s}, f_{t}) vol(f^{∗}g) dt = ∫_{a}^{b} ∫_{M}(∂_{t}g(f_{s}, f_{t}) − g(f_{s}, ∇_{∂_{t}}^{g}f_{t})) vol(f^{∗}g) dt = − ∫_{a}^{b} ∫_{M} g(f_{s}, f_{t})∂_{t} vol(f^{∗}g) dt − ∫_{a}^{b} ∫_{M} g(f_{s}, ∇_{∂_{t}}^{g}f_{t}) vol(f^{∗}g) dt = ∫_{a}^{b} ∫_{M}(−g(f_{s}, f_{t}) div^{f^{∗}g}(f_{t}^{⊤}) + g(f_{s}, f_{t})g(f_{t}^{⊥}, Tr^{f^{∗}g}(S^{f}))−

− g(f_{s}, ∇_{∂_{t}}^{g}f_{t})) vol(f^{∗}g) dt

The second summand yields:

∫_{a}^{b} ∫_{M} \frac{1}{2}∥f_{t}∥_{g}^{2} div^{f^{∗}g}(f_{s}^{⊤}) vol(f^{∗}g) dt

= − ∫_{a}^{b} ∫_{M} \frac{1}{2}(f^{∗}g)(f_{s}^{⊤}, grad^{f^{∗}g}(∥f_{t}∥_{g}^{2})) vol(f^{∗}g) dt = − ∫_{a}^{b} ∫_{M} \frac{1}{2}g(f_{s}, T f. grad^{f^{∗}g}(∥f_{t}∥_{g}^{2})) vol(f^{∗}g) dt

Documenta Mathematica 10 (2005) 217–245

Thus the first variation ∂_{s}E_{G^{0}}(f) is:

∫_{a}^{b} ∫_{M} g(f_{s}, − ∇_{∂_{t}}^{g}f_{t} − div^{f^{∗}g}(f_{t}^{⊤})f_{t} + g(f_{t}^{⊥}, Tr^{f^{∗}g}(S^{f}))f_{t}

− \frac{1}{2}T f. grad^{f^{∗}g}(∥f_{t}∥_{g}^{2}) − \frac{1}{2}∥f_{t}∥_{g}^{2}g(f_{s}^{⊥}, Tr^{f^{∗}g}(S^{f}))) vol(f^{∗}g) dt. □

4.2. Geodesics for G^{0} in B_{i}(M, N). We restrict to geodesics t ↦ f(t, ) in Imm(M, N) which are horizontal: g(f_{t}, T f) = 0. Then f_{t}^{⊤} = 0 and f_{t} = f_{t}^{⊥}, so equation 4.1.(1) becomes

∇_{∂_{t}}^{g}f_{t} − g(f_{t}, Tr^{f^{∗}g}(S^{f}))f_{t} + \frac{1}{2}T f. grad^{f^{∗}g}(∥f_{t}∥_{g}^{2}) + \frac{1}{2}∥f_{t}∥_{g}^{2} Tr^{f^{∗}g}(S^{f}) = 0.

It splits into a vertical (tangential) part

−T f.(∇_{∂_{t}}f_{t})^{⊤} + \frac{1}{2}T f. grad^{f^{∗}g}(∥f_{t}∥_{g}^{2}) = 0

which vanishes identically since

(f^{∗}g)(grad^{f^{∗}g}(∥f_{t}∥_{g}^{2}), X) = X(g(f_{t}, f_{t})) = 2g(∇_{X}f_{t}, f_{t}) = 2g(∇_{∂_{t}}^{g}T f.X, f_{t})

= 2∂_{t}g(T f.X, f_{t}) − 2g(T f.X, ∇_{∂_{t}}^{g}f_{t}) = −2g(T f.X, ∇_{∂_{t}}^{g}f_{t}),

and a horizontal (normal) part which is the geodesic equation in B_{i}:

(1) ∇_{∂_{t}}^{N(f)}f_{t} − g(f_{t}, Tr^{f^{∗}g}(S^{f}))f_{t} + \frac{1}{2}∥f_{t}∥_{g}^{2} Tr^{f^{∗}g}(S^{f}) = 0, g(T f, f_{t}) = 0

4.3. The induced metric of G^{0} in B_{i}(M, N) in a chart. Let f_{0} : M → N be a fixed immersion which will be the ‘center’ of our chart. Let N(f_{0}) ⊂ f_{0}^{∗}T N be the normal bundle to f_{0}. Let exp^{g} : N(f_{0}) → N be the exponential map for the metric g and let V ⊂ N(f_{0}) be a neighborhood of the 0 section on which the exponential map is an immersion. Consider the mapping

ψ = ψ_{f_{0}} : Γ(V) → Imm(M, N), , ψ(Γ(V)) =: Q(f_{0}), (1)

ψ(a)(x) = exp^{g}(a(x)) = exp_{f_{0}(x)}^{g}(a(x)).

The inverse (on its image) of π ∘ ψ_{f} : Γ(V) → B_{i}(M, N) is a smooth chart on B_{i}(M, N). Our goal is to calculate the induced metric on this chart, that is

((π ∘ ψ_{f_{0}})^{∗}G_{a}^{0})(b_{1}, b_{2})

for any a ∈ Γ(V), b_{1}, b_{2} ∈ Γ(N(f_{0})). This will enable us to calculate the sectional curvatures of B_{i}.

We shall fix the section a and work with the ray of points t.a in this chart. Everything will revolve around the map:

f(t, x) = ψ(t.a)(x) = exp^{g}(t.a(x)).

We shall also use a fixed chart (M ⊃ U ⟶^{u} ℝ^{m}) on M with ∂_{i} = ∂/∂u^{i}. Then x ↦ (t ↦ f(t, x)) = exp_{f_{0}(x)}^{g}(t.a(x)) is a variation consisting entirely of geodesics, thus:

t ↦ ∂_{i}f(t, x) = T f.∂_{i} =: Z_{i}(t, x, a) is the Jacobi field along t ↦ f(t, x)with
(2)  Z_{i}(0, x, a) = ∂_{i}|_{x} exp_{f_{0}(x)}(0) = ∂_{i}|_{x}f_{0} = T_{x}f_{0}.∂_{i}|_{x}, and
(∇_{∂_{t}}^{g}Z_{i})(0, x) = (∇_{∂_{t}}^{g}T f.∂_{i})(0, x) = (∇_{∂_{i}}^{g}T f.∂_{t})(0, x) =
= ∇_{∂_{i}}^{g}(∂_{t}|_{0} exp_{f_{0}(x)}(t.a(x))) = (∇_{∂_{i}}^{g}a)(x).

Then the pullback metric is given by

f^{∗}g = ψ(ta)^{∗}g = g(T f, T f) =

= ∑_{i,j=1}^{m} g(T f.∂_{i}, T f, ∂_{j}) du^{i} ⊗ du^{j} = ∑_{i,j=1}^{m} g(Z_{i}, Z_{j}) du^{i} ⊗ du^{j}. (3)

The induced volume density is:

(4)  vol(f^{∗}g) = \sqrt{det(g(Z_{i}, Z_{j}))}|du^{1} ∧ ⋯ ∧ du^{m}|

Moreover we have for a ∈ Γ(V) and b ∈ Γ(N(f_{0}))

(T_{ta}ψ.tb)(x) = ∂_{s}|_{0} exp_{f_{0}(x)}^{g}(ta(x) + stb(x)))

(5) = Y (t, x, a, b) for the Jacobi field Y along t ↦ f(t, x) with

Y (0, x, a, b) = 0_{f_{0}(x)},
(∇_{∂_{t}}^{g}Y ( , x, a, b))(0) = ∇_{∂_{t}}^{g}∂_{s} exp_{f_{0}(x)}^{g}(ta(x) + stb(x))
= ∇_{∂_{s}}^{g}∂_{t}|_{0} exp_{f_{0}(x)}^{g}(ta(x) + stb(x))
= ∇_{∂_{s}}^{g}(a(x) + s.b(x))|_{s=0} = b(x).

Now we want to split T_{a}ψ.b into vertical (tangential) and horizontal parts with respect to the immersion ψ(ta) = f(t, ). The tangential part has locally the form

T f.(T_{ta}ψ.tb)^{⊤} = ∑_{i=1}^{m} c^{i} T f.∂_{i} = ∑_{i=1}^{m} c^{i} Z_{i} where for all j

g(Y, Z_{j}) = g (∑_{i=1}^{m} c^{i} Z_{i}, Z_{j}) = ∑_{i=1}^{m} c^{i} (f^{∗}g)_{ij},

c^{i} = ∑_{j=1}^{m}(f^{∗}g)^{ij}g(Y, Z_{j}).

Thus the horizontal part is

(T_{ta}ψ.tb)^{⊥} = Y^{⊥} = Y − ∑_{i=1}^{m} c^{i} Z_{i} = Y − ∑_{i,j=1}^{m}(f^{∗}g)^{ij} g(Y, Z_{j}) Z_{i} (6)

Thus the induced metric on B_{i}(M, N) has the following expression in the chart (π ∘ ψ_{f_{0}})^{−1}, where a ∈ Γ(V) and b_{1}, b_{2} ∈ Γ(N(f_{0})):

((π ∘ ψ_{f_{0}})^{∗}G^{0})_{ta}(b_{1}, b_{2}) = G_{π(ψ(ta))}^{0}(T_{ta}(π ∘ ψ)b_{1}, T_{ta}(π ∘ ψ)b_{2})
= G_{ψ(ta)}^{0}((T_{ta}ψ.b_{1})^{⊥}, (T_{ta}ψ.b_{2})^{⊥})
= ∫_{M} g((T_{ta}ψ.b_{1})^{⊥}, (T_{ta}ψ.b_{2})^{⊥}) vol(f^{∗}g)

= ∫_{M} \frac{1}{t^{2}} g(Y (b_{1}) − ∑_{i,j}(f^{∗}g)^{ij} g(Y (b_{1}), Z_{j}) Z_{i} , Y (b_{2})) (7)

\sqrt{det(g(Z_{i}, Z_{j}))}|du^{1} ∧ ⋯ ∧ du^{m}|

4.4. Expansion to order 2 of the induced metric of G^{0} in B_{i}(M, N) in a chart. We use the setting of 4.3, the Einstein summation convention, and the abbreviations f_{i} := ∂_{i}f_{0} = ∂_{u^{i}}f_{0} and ∇_{i}^{g} := ∇_{∂_{i}}^{g} = ∇_{∂_{u^{i}}}^{g}. We compute the expansion in t up to order 2 of the metric 4.3.(7). Our method is to use the Jacobi equation

∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Y = R^{g}(ċ, Y)ċ

which holds for any Jacobi field Y along a geodesic c. By 2.6.(2) we have:

(1) g(∇_{i}^{g}a, f_{j}) = −(f_{0}^{∗}g)(L_{a}^{f_{0}}(∂_{i}), ∂_{j}) = −g(a, S^{f_{0}}(f_{i}, f_{j})) = g(a, S_{ij}^{f_{0}})

We start by expanding the pullback metric 4.3.(3) and its inverse:

∂_{t}(f^{∗}g)_{ij} = ∂_{t}g(Z_{i}, Z_{j}) = g(∇_{∂_{t}}^{g}Z_{i}, Z_{j}) + g(Z_{i}, ∇_{∂_{t}}^{g}Z_{j})
∂_{t}^{2}g(Z_{i}, Z_{j}) = g(∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Z_{i}, Z_{j}) + 2g(∇_{∂_{t}}^{g}Z_{i}, ∇_{∂_{t}}^{g}Z_{j}) + g(Z_{i}, ∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Z_{j})
(f^{∗}g)_{ij} = (f_{0}^{∗}g)_{ij} + t(g(∇_{i}^{g}a, f_{j}) + g(f_{i}, ∇_{j}^{g}a))+
+ \frac{1}{2}t^{2}(g(R^{g}(a, f_{i})a, f_{j}) + 2g(∇_{i}^{g}a, ∇_{j}^{g}a) + g(f_{i}, R^{g}(a, f_{j})a))
+ O(t^{3})
= (f_{0}^{∗}g)_{ij} − 2t(f_{0}^{∗}g)(L_{a}^{f_{0}}(∂_{i}), ∂_{j})

+ t^{2}(g(R^{g}(a, f_{i})a, f_{j}) + g(∇_{i}^{g}a, ∇_{j}^{g}a)) + O(t^{3}) (2)

We expand now the volume form vol(f^{∗}g) = \sqrt{det(g(Z_{i}, Z_{j}))}|du^{1} ∧ ⋯ ∧ du^{m}|. The time derivative at 0 of the inverse of the pullback metric is:

∂_{t}(f^{∗}g)^{ij}|_{0} = −(f_{0}^{∗}g)^{ik}(∂_{t}|_{0}(f^{∗}g)_{kl})(f_{0}^{∗}g)^{lj} = −(f_{0}^{∗}g)^{ik}(f_{0}^{∗}g)(a, S_{kl}^{f_{0}})(f_{0}^{∗}g)^{lj}

Therefore,

∂_{t}\sqrt{det(g(Z_{i}, Z_{j}))} = \frac{1}{2}(f^{∗}g)^{ij}∂_{t}(g(Z_{i}, Z_{j}))\sqrt{det(g(Z_{i}, Z_{j}))}
∂_{t}^{2}\sqrt{det(g(Z_{i}, Z_{j}))} = \frac{1}{2}∂_{t}(f^{∗}g)^{ij}∂_{t}(g(Z_{i}, Z_{j}))\sqrt{det(g(Z_{i}, Z_{j}))}
+ \frac{1}{2}(f^{∗}g)^{ij}∂_{t}^{2}(g(Z_{i}, Z_{j}))\sqrt{det(g(Z_{i}, Z_{j}))}

Documenta Mathematica 10 (2005) 217–245

+ \frac{1}{2}(f^{∗}g)^{ij}∂_{t}(g(Z_{i}, Z_{j}))∂_{t}\sqrt{det(g(Z_{i}, Z_{j}))}

and so

vol(f^{∗}g) = \sqrt{det(g(Z_{i}, Z_{j}))}|du^{1} ∧ ⋯ ∧ du^{m}|
= (1 − t Tr(L_{a}^{f_{0}}) + t^{2}(− Tr(L_{a}^{f_{0}} ∘ L_{a}^{f_{0}}) + \frac{1}{2}(Tr(L_{a}^{f_{0}}))^{2}

+ \frac{1}{2}(f^{∗}g)^{ij}(g(R^{g}(a, f_{i})a, f_{j}) + g(∇_{i}^{g}a, ∇_{j}^{g}a))) + O(t^{3})) vol(f_{0}^{∗}g). (3)

Moreover, by 2.6.(2) we may split ∇_{i}^{g}a = −T f_{0}.L_{a}^{f_{0}}(∂_{i})+∇_{i}^{N(f_{0})}a and we write ∇_{i}^{⊥}a for ∇_{i}^{N(f_{0})}a shortly. Thus:

(f_{0}^{∗}g)^{ij}g(∇_{i}^{g}a, ∇_{j}^{g}a) = (f_{0}^{∗}g)^{ij} (g(T f_{0}.L_{a}^{f_{0}}(∂_{i}), T f_{0}.L_{a}^{f_{0}}(∂_{j})) + g(∇_{i}^{⊥}a, ∇_{j}^{⊥}a))
= Tr(L_{a}^{f_{0}} ∘ L_{a}^{f_{0}}) + (f_{0}^{∗}g)^{ij}g(∇_{i}^{⊥}a, ∇_{j}^{⊥}a)

and so that the tangential term above combines with the first t^{2} term in the expansion of the volume, changing its coefficient from −1 to −\frac{1}{2}. Let us now expand

g((T_{ta}ψ.tb_{1})^{⊥},(T_{ta}ψ.tb_{2})^{⊥})
= g(Y (b_{1}) − (f^{∗}g)^{ij}g(Y (b_{1}), Z_{j}) Z_{i} , Y (b_{2}))
= g(Y (b_{1}), Y (b_{2})) − (f^{∗}g)^{ij}g(Y (b_{1}), Z_{j})g(Z_{i}, Y (b_{2})).

We have:

∂_{t}g(Y (b_{1}), Y (b_{2})) = g(∇_{∂_{t}}^{g}Y (b_{1}), Y (b_{2})) + g(Y (b_{1}), ∇_{∂_{t}}^{g}Y (b_{2}))
∂_{t}^{2}g(Y (b_{1}), Y (b_{2})) = g(∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Y (b_{1}), Y (b_{2})) + 2g(∇_{∂_{t}}^{g}Y (b_{1}), ∇_{∂_{t}}^{g}Y (b_{2}))
+ g(Y (b_{1}), ∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Y (b_{2}))
= 2g(R^{g}(a, Y (b_{1}))a, Y (b_{2})) + 2g(∇_{∂_{t}}^{g}Y (b_{1}), ∇_{∂_{t}}^{g}Y (b_{2}))
∂_{t}g(Y (b_{1}), Z_{j}) = g(∇_{∂_{t}}^{g}Y (b_{1}), Z_{j}) + g(Y (b_{1}), ∇_{∂_{t}}^{g}Z_{j})
∂_{t}^{2}g(Y (b_{1}), Z_{j}) = 2g(R^{g}(a, Y (b_{1}))a, Z_{j}) + 2g(∇_{∂_{t}}^{g}Y (b_{1}), ∇_{∂_{t}}^{g}Z_{j})

Note that: Y (0, h) = 0, (∇_{∂_{t}}^{g}Y (h))(0) = h, (∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Y (h))(0) = R^{g}(a, Y (0, h))a = 0, (∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}∇_{∂_{t}}^{g}Y (h))(0) = R^{g}(a, ∇_{∂_{t}}^{g}Y (h)(0))a = R^{g}(a, h)a. Thus: g(Y (b_{1}), Y (b_{2})) − (f^{∗}g)^{ij}g(Y (b_{1}), Z_{j})g(Z_{i}, Y (b_{2})) = t^{2}g(b_{1}, b_{2}) + t^{4}(\frac{1}{3}g(R^{g}(a, b_{1})a, b_{2}) − (f_{0}^{∗}g)^{ij}g(b_{1}, ∇_{j}^{⊥}a)g(∇_{i}^{⊥}a, b_{2})) + O(t^{5}). The expansion of G^{0} up to order 2 is thus:

((π ∘ ψ_{f_{0}})^{∗}G^{0})_{ta}(b_{1}, b_{2}) =

= ∫_{M} \frac{1}{t^{2}} g(Y (b_{1}) − ∑_{i,j}(f^{∗}g)^{ij} g(Y (b_{1}), Z_{j}) Z_{i} , Y (b_{2})) vol(f^{∗}g)

= ∫_{M}(g(b_{1}, b_{2}) vol(f_{0}^{∗}g) − t ∫_{M} g(b_{1}, b_{2}) Tr(L_{a}^{f_{0}}) vol(f_{0}^{∗}g)
+ t^{2} ∫_{M}(g(b_{1}, b_{2})(−\frac{1}{2} Tr(L_{a}^{f_{0}} ∘ L_{a}^{f_{0}}) + \frac{1}{2} Tr(L_{a}^{f_{0}})^{2}
+ \frac{1}{2}(f^{∗}g)^{ij}g(R^{g}(a, f_{i})a, f_{j}) + \frac{1}{2}(f^{∗}g)^{ij}g(∇_{i}^{⊥}a, ∇_{j}^{⊥}a))
+ \frac{1}{3}g(R^{g}(a, b_{1})a, b_{2}) − (f_{0}^{∗}g)^{ij}g(b_{1}, ∇_{j}^{⊥}a)g(∇_{i}^{⊥}a, b_{2})) vol(f_{0}^{∗}g)

+ O(t^{3}) (4)

4.5. Computation of the sectional curvature in B_{i}(M, N) at f_{0}. We use the following formula which is valid in a chart:

2R_{a}(m, h, m, h) = 2G_{a}^{0}(R_{a}(m, h)m, h) =

= −2d^{2}G^{0}(a)(m, h)(h, m) + d^{2}G^{0}(a)(m, m)(h, h) + d^{2}G^{0}(a)(h, h)(m, m)

− 2G^{0}(Γ(h, m), Γ(m, h)) + 2G^{0}(Γ(m, m), Γ(h, h))

The sectional curvature at the two-dimensional subspace P_{a}(m, h) of the tangent space which is spanned by m and h is then given by:

k_{a}(P (m, h)) = −\frac{G_{a}^{0}(R(m, h)m, h)}{∥m∥^{2}∥h∥^{2} − G_{a}^{0}(m, h)^{2}} .

We compute this directly for a = 0. From the expansion up to order 2 of G_{ta}^{0}(b_{1}, b_{2}) in 4.4.(4) we get

dG^{0}(0)(a)(b_{1}, b_{2}) = − ∫_{M} g(b_{1}, b_{2})g(a, Tr^{f_{0}^{∗}g} S^{f_{0}}) vol(f_{0}^{∗}g)

and we compute the Christoffel symbol:

−2G_{0}^{0}(Γ_{0}(a, b), c) = −dG^{0}(0)(c)(a, b) + dG^{0}(0)(a)(b, c) + dG^{0}(0)(b)(c, a)
= ∫_{M}(g(a, b)g(c, Tr^{f_{0}^{∗}g}(S^{f_{0}})) − g(b, c)g(a, Tr^{f_{0}^{∗}g}(S^{f_{0}}))
− g(c, a)g(b, Tr^{f_{0}^{∗}g}(S^{f_{0}}))) vol(f_{0}^{∗}g)
= ∫_{M} g(c, g(a, b) Tr^{f_{0}^{∗}g}(S^{f_{0}}) − Tr(L_{a}^{f_{0}})b − Tr(L_{b}^{f_{0}})a) vol(f_{0}^{∗}g)
Γ_{0}(a, b) = −\frac{1}{2}g(a, b) Tr^{f_{0}^{∗}g}(S^{f_{0}}) + \frac{1}{2} Tr(L_{a}^{f_{0}})b + \frac{1}{2} Tr(L_{b}^{f_{0}})a

The expansion 4.4.(4) also gives:

\frac{1}{2!}d^{2}G_{0}^{0}(a_{1}, a_{2})(b_{1}, b_{2}) =

= ∫_{M}(g(b_{1}, b_{2})(−\frac{1}{2} Tr(L_{a_{1}}^{f_{0}} ∘ L_{a_{2}}^{f_{0}}) + \frac{1}{2} Tr(L_{a_{1}}^{f_{0}}) Tr(L_{a_{2}}^{f_{0}})

+ \frac{1}{2}(f^{∗}g)^{ij}g(R^{g}(a_{1}, f_{i})a_{2}, f_{j}) + \frac{1}{2}(f^{∗}g)^{ij}g(∇_{i}^{⊥}a_{1}, ∇_{j}^{⊥}a_{2}))

+ \frac{1}{6}g(R^{g}(a_{1}, b_{1})a_{2}, b_{2}) + \frac{1}{6}g(R^{g}(a_{2}, b_{1})a_{1}, b_{2})

− \frac{1}{2}(f_{0}^{∗}g)^{ij}g(b_{1}, ∇_{j}^{⊥}a_{1})g(∇_{i}^{⊥}a_{2}, b_{2})
− \frac{1}{2}(f_{0}^{∗}g)^{ij}g(b_{1}, ∇_{j}^{⊥}a_{2})g(∇_{i}^{⊥}a_{1}, b_{2})) vol(f_{0}^{∗}g) + O(t^{3})

Thus we have:

− d^{2}G^{0}(0)(x, y)(y, x) + \frac{1}{2}d^{2}G^{0}(0)(x, x)(y, y) + \frac{1}{2}d^{2}G^{0}(0)(y, y)(x, x) =

= ∫_{M}(−2g(y, x)(−\frac{1}{2} Tr(L_{x}^{f_{0}} ∘ L_{y}^{f_{0}}) + \frac{1}{2} Tr(L_{x}^{f_{0}}) Tr(L_{y}^{f_{0}})

+ \frac{1}{2}(f_{0}^{∗}g)^{ij}g(R^{g}(x, f_{i})y, f_{j}) + \frac{1}{2}(f_{0}^{∗}g)^{ij}g(∇_{i}^{⊥}x, ∇_{j}^{⊥}y))

+ g(y, y)(−\frac{1}{2} Tr(L_{x}^{f_{0}} ∘ L_{x}^{f_{0}}) + \frac{1}{2} Tr(L_{x}^{f_{0}}) Tr(L_{x}^{f_{0}})

+ \frac{1}{2}(f_{0}^{∗}g)^{ij}g(R^{g}(x, f_{i})x, f_{j}) + \frac{1}{2}(f_{0}^{∗}g)^{ij}g(∇_{i}^{⊥}x, ∇_{j}^{⊥}x))

+ g(x, x)(−\frac{1}{2} Tr(L_{y}^{f_{0}} ∘ L_{y}^{f_{0}}) + \frac{1}{2} Tr(L_{y}^{f_{0}}) Tr(L_{y}^{f_{0}})

+ \frac{1}{2}(f_{0}^{∗}g)^{ij}g(R^{g}(y, f_{i})y, f_{j}) + \frac{1}{2}(f_{0}^{∗}g)^{ij}g(∇_{i}^{⊥}y, ∇_{j}^{⊥}y))

+ g(R^{g}(y, x)y, x)
+ (f_{0}^{∗}g)^{ij} (g(y, ∇_{j}^{⊥}y)g(∇_{i}^{⊥}x, x) + g(y, ∇_{j}^{⊥}x)g(∇_{i}^{⊥}y, x))

− (f_{0}^{∗}g)^{ij} (g(x, ∇_{j}^{⊥}y)g(∇_{i}^{⊥}y, x) + g(y, ∇_{j}^{⊥}x)g(∇_{i}^{⊥}x, y))) vol(f_{0}^{∗}g)

For the second part of the curvature we have

−G_{0}(Γ_{0}(x, y),Γ_{0}(y, x)) + G_{0}(Γ_{0}(y, y), Γ_{0}(x, x)) =
= \frac{1}{4}∫_{M}((∥x∥_{g}^{2}∥y∥_{g}^{2} − g(x, y)^{2})∥ Tr^{f_{0}^{∗}g}(S^{f})∥_{g}^{2}
− 3∥ Tr(L_{x}^{f_{0}})y − Tr(L_{y}^{f_{0}})x∥_{g}^{2}) vol(f_{0}^{∗}g)

To organize all these terms in the curvature tensor, note that they belong to three types: terms which involve the second fundamental form L^{f_{0}}, terms which involve the curvature tensor R^{g} of N and terms which involve the normal component of the covariant derivative ∇^{⊥}a. There are 3 of the first type, two of the second and the ones of the third can be organized neatly into two also. The final curvature tensor is the integral over M of their sum. Here are the terms in detail:

(1) Terms involving the trace of products of L’s. These are:

−\frac{1}{2}(g(y, y) Tr(L_{x}^{f_{0}} ∘ L_{x}^{f_{0}}) − 2g(x, y) Tr(L_{x}^{f_{0}} ∘ L_{y}^{f_{0}}) + g(x, x) Tr(L_{y}^{f_{0}} ∘ L_{y}^{f_{0}})).

Note that x and y are sections of the normal bundle N(f_{0}), so we may define x ∧ y to be the induced section of ⋀^{2} N(f_{0}). Then the expression inside the parentheses is a positive semi-definite quadratic function of x ∧ y. To see this, note a simple linear algebra fact – that if Q(a, b) is any positive semi-definite inner product on ℝ^{n}, then

Q̃(a ∧ b, c ∧ d) =
< a, c >Q(b, d)− < a, d > Q(b, c)+ < b, d > Q(a, c)− < b, c > Q(a, d)
Q̃(a ∧ b, a ∧ b) = ∥a∥^{2}Q(b, b) − 2 < a, b > Q(a, b) + ∥b∥^{2}Q(a, a)

is a positive semi-definite inner product on ⋀^{2} V. In particular, Tr(L_{x}^{f_{0}} ∘ L_{y}^{f_{0}}) is a positive semi-definite inner product on the normal bundle, hence it defines a positive semi-definite inner product T̃r(L^{f_{0}} ∘ L^{f_{0}}) on ⋀^{2} N(f_{0}). Thus:

term(1) = −\frac{1}{2} T̃r(L^{f_{0}} ∘ L^{f_{0}})(x ∧ y) ≤ 0.

(2) Terms involving trace of one L. We have terms both from the second and first derivatives of G, namely

\frac{1}{2}(g(y, y) Tr(L_{x}^{f_{0}})^{2} − 2g(x, y) Tr(L_{x}^{f_{0}}) Tr(L_{y}^{f_{0}}) + g(x, x) Tr(L_{y}^{f_{0}})^{2})

and

−\frac{3}{4}∥ Tr(L_{x}^{f_{0}})y − Tr(L_{y}^{f_{0}})x∥_{g}^{2}

which are the same up to their coefficients. Their sum is

term(2) = −\frac{1}{4}∥ Tr(L_{x}^{f_{0}})y − Tr(L_{y}^{f_{0}})x∥_{g}^{2} ≤ 0.

Note that this is a function of x ∧ y also.

(3) The term involving the norm of the second fundamental form. Since ∥x∥_{g}^{2}∥y∥_{g}^{2} − g(x, y)^{2} = ∥x ∧ y∥_{g}^{2}, this term is just:

term(3) = +\frac{1}{4}∥x ∧ y∥_{g}^{2}∥ Tr^{g}(S^{f_{0}})∥_{g}^{2} ≥ 0.

(4) The curvature of N term. This is

term(4) = g(R^{g}(x, y)x, y).

Note that because of the skew-symmetry of the Riemann tensor, this is a function of x ∧ y also.

(5) The Ricci-curvature-like term. The other curvature terms are

\frac{1}{2}(f_{0}^{∗}g)^{ij}(g(x, x)g(R^{g}(y, f_{i})y, f_{j})
− 2g(x, y)g(R^{g}(x, f_{i})y, f_{j}) + g(y, y)g(R^{g}(x, f_{i})x, f_{j})).

If V and W are two perpendicular subspaces of the tangent space T N_{p} at a point p, then we can define a ‘cross Ricci curvature’ Ric(V, W) in terms of bases {v_{i}}, {w_{j}} of V and W by

Ric(V, W) = g^{ij}g^{kl}g(R^{g}(v_{i}, w_{k})v_{j}, w_{l}).

Then this term factors as

term(5) = ∥x ∧ y∥_{g}^{2} Ric(T M, span(x, y)).

(6 -7) Terms involving the covariant derivative of a. It is remarkable that, so far, every term in the curvature tensor of B_{i} vanishes if x ∧ y ≡ 0, e.g., if the codimension of N in M is one! Now we have the terms

(f_{0}^{∗}g)^{ij}(g(x, y)g(∇_{i}^{⊥}x, ∇_{j}^{⊥}y) − \frac{1}{2}g(x, x)g(∇_{i}^{⊥}y, ∇_{j}^{⊥}y) − \frac{1}{2}g(y, y)g(∇_{i}^{⊥}x, ∇_{j}^{⊥}x)
− g(x, ∇_{i}^{⊥}x)g(y, ∇_{j}^{⊥}y) − g(x, ∇_{i}^{⊥}y)g(y, ∇_{j}^{⊥}x)
+ g(x, ∇_{i}^{⊥}y)g(x, ∇_{j}^{⊥}y) + g(y, ∇_{i}^{⊥}x)g(y, ∇_{j}^{⊥}x)).

To understand this expression, we need a linear algebra computation, namely that if a, b, a′, b′ ∈ ℝ^{n}, then:

< a, b >< a′, b′ > − < a, a′ >< b, b′ > − < a, b′ >< b, a′ > −

− \frac{1}{2} < a, a >< b′, b′ > −\frac{1}{2} < b, b >< a′, a′ > + < a, b′ >^{2} + < b, a′ >^{2}=

= \frac{1}{2}(< a, b′ > − < b, a′ >)^{2} − \frac{1}{2}∥a ∧ b′ − b ∧ a′∥^{2}

Note that the term g(x, ∇^{⊥}y) (without an i) is a section of Ω_{M}^{1} and the sum over i and j is just the norm in Ω_{M}^{1}, so the above computation applies and the expression splits into 2 terms:

term(6) = −\frac{1}{2}∥(g(x, ∇^{⊥}y) − g(y, ∇^{⊥}x)∥_{Ω_{M}^{1}}^{2} ≤ 0
term(7) = \frac{1}{2}∥x ∧ ∇^{⊥}y − y ∧ ∇^{⊥}x∥_{Ω_{M}^{1} ⊗∧^{2}N(f)}^{2} ≥ 0

Altogether, we get that the Riemann curvature of B_{i} is the integral over M of the sum of the above 7 terms. We have the Corollary:

Corollary. If the codimension of M in N is one, then all sectional curvatures of B_{i} are non-negative. For any codimension, sectional curvature in the plane spanned by x and y is non-negative if x and y are parallel, i.e., x ∧ y = 0 in ⋀^{2} T^{∗}N.

In general, the negative terms in the curvature tensor (giving positive sectional curvature) are clearly connected with the vanishing of geodesic distance: in some directions the space wraps up on itself in tighter and tighter ways. However, in codimension two or more with a flat ambient space N (so terms (4) and (5) vanish), there seem to exist conflicting tendencies making B_{i} close up or open up: terms (1), (2) and (6) give positive curvature, while terms (3) and (7) give negative curvature. It would be interesting to explore the geometrical meaning of these, e.g., for manifolds of space curves.

5. Vanishing geodesic distance on groups of diffeomorphisms

5.1. The H^{0}-metric on groups of diffeomorphisms. Let (N, g) be a smooth connected Riemannian manifold, and let Diff_{c}(N) be the group of all diffeomorphisms with compact support on N, and let Diff_{0}(N) be the subgroup of those which are diffeotopic in Diff_{c}(N) to the identity; this is the connected component of the identity in Diff_{c}(N), which is a regular Lie group in the sense of [8], section 38, see [8], section 42. The Lie algebra is 𝔛_{c}(N), the space of all smooth vector fields with compact support on N, with the negative of the usual bracket of vector fields as Lie bracket. Moreover, Diff_{0}(N) is a simple group (has no nontrivial normal subgroups), see [5], [14], [9]. The right invariant H^{0}- metric on Diff_{0}(N) is then given as follows, where h, k : N → T N are vector fields with compact support along φ and where X = h ∘ φ^{−1}, Y = k ∘ φ^{−1} ∈ 𝔛_{c}(N):

G_{φ}^{0}(h, k) = ∫_{N} g(h, k) vol(φ^{∗}g) = ∫_{N} g(X ∘ φ, Y ∘ φ)φ^{∗} vol(g)

= ∫_{N} g(X, Y) vol(g) (1)

5.2. Theorem. Geodesic distance on Diff_{0}(N) with respect to the H^{0}-metric vanishes.

Proof. Let [0, 1] ∋ t ↦ φ(t, ) be a smooth curve in Diff_{0}(N) between φ_{0} and φ_{1}. Consider the curve u = φ_{t} ∘ φ^{−1} in 𝔛_{c}(N), the right logarithmic derivative. Then for the length and the energy we have:

(1)  L_{G^{0}}(φ) = ∫_{0}^{1} \sqrt{∫_{N} ∥u∥_{g}^{2} vol(g)} dt

(2)  E_{G^{0}}(φ) = ∫_{0}^{1} ∫_{N} ∥u∥_{g}^{2} vol(g) dt
(3)  L_{G^{0}}(φ)^{2} ≤ E_{G^{0}}(φ)

(4) Let us denote by Diff_{0}(N)^{E=0} the set of all diffeomorphisms φ ∈ Diff_{0}(N) with the following property: For each ε > 0 there exists a smooth curve from the identity to φ in Diff_{0}(N) with energy ≤ ε.

(5) We claim that Diff_{0}(N)^{E=0} coincides with the set of all diffeomorphisms which can by reached from the identity by a smooth curve of arbitraily short G^{0}-length. This follows by (3).

(6) We claim that Diff_{0}(N)^{E=0} is a normal subgroup of Diff_{0}(N). Let φ_{1} ∈ Diff_{0}(N)^{E=0} and ψ ∈ Diff_{0}(N). For any smooth curve t ↦ φ(t, ) from the identity to φ_{1} with energy E_{G^{0}}(φ) < ε we have

E_{G^{0}}(ψ^{−1} ∘ φ ∘ ψ) = ∫_{0}^{1} ∫_{N} ∥T ψ^{−1} ∘ φ_{t} ∘ ψ∥_{g}^{2} vol((ψ^{−1} ∘ φ ∘ ψ)^{∗}g)

≤ sup_{x∈N} ∥T_{x}ψ^{−1}∥^{2} · ∫_{0}^{1} ∫_{N} ∥φ_{t} ∘ ψ∥_{g}^{2}(φ ∘ ψ)^{∗} vol((ψ^{−1})^{∗}g)

≤ sup_{x∈N} ∥T_{x}ψ^{−1}∥^{2} · sup_{x∈N} \frac{vol((ψ^{−1})^{∗}g)}{vol(g)} · ∫_{0}^{1} ∫_{N} ∥φ_{t} ∘ ψ∥_{g}^{2} (φ ∘ ψ)^{∗} vol(g)

≤ sup_{x∈N} ∥T_{x}ψ^{−1}∥^{2} · sup_{x∈N} \frac{vol((ψ^{−1})^{∗}g)}{vol(g)} · E_{G^{0}}(φ).

Since ψ is a diffeomorphism with compact support, the two suprema are bounded. Thus ψ^{−1} ∘ φ_{1} ∘ ψ ∈ Diff_{0}(N)^{E=0}.

(7) We claim that Diff_{0}(N)^{E=0} is a non-trivial subgroup. In view of the simplicity of Diff_{0}(N) mentioned in 5.1 this concludes the proof.

It remains to find a non-trivial diffeomorphism in Diff_{0}(N)^{E=0}. The idea is to use compression waves. The basic case is this: take any non-decreasing smooth function f : ℝ → ℝ such that f(x) ≡ 0 if x ≪ 0 and f(x) ≡ 1 if x ≫ 0. Define

φ(t, x) = x + f(t − λx)

where λ < 1/ max(f′). Note that

φ_{x}(t, x) = 1 − λf′(t − λx) > 0,

hence each map φ(t, ) is a diffeomorphism of ℝ and we have a path in the group of diffeomorphisms of ℝ. These maps are not the identity outside a compact set however. In fact, φ(x) = x + 1 if x ≪ 0 and φ(x) = x if x ≫ 0. As t → −∞, the map φ(t, ) approaches the identity, while as t → +∞, the map approaches translation by 1. This path is a moving compression wave which pushes all points forward by a distance 1 as it passes. We calculate its energy between two times t_{0} and t_{1}:

E_{t_{0}}^{t_{1}}(φ) = ∫_{t_{0}}^{t_{1}} ∫_{ℝ} φ_{t}(t, φ(t, )^{−1}(x))^{2}dx dt = ∫_{t_{0}}^{t_{1}} ∫_{ℝ} φ_{t}(t, y)^{2}φ_{y}(t, y)dy dt

= ∫_{t_{0}}^{t_{1}} ∫_{ℝ} f′(z)^{2} · (1 − λf′(z))dz/λ dt

≤ \frac{max f^{′2}}{λ} · (t_{1} − t_{0}) · ∫_{supp(f′)}(1 − λf′(z))dz

If we let λ = 1 − ε and consider the specific f given by the convolution

f(z) = max(0, min(1, z)) ⋆ G_{ε}(z),

where G_{ε} is a smoothing kernel supported on [−ε, +ε], then the integral is bounded by 3ε, hence

E_{t_{0}}^{t_{1}}(φ) ≤ (t_{1} − t_{0})\frac{3ε}{1−ε}.

We next need to adapt this path so that it has compact support. To do this we have to start and stop the compression wave, which we do by giving it variable length. Let:

f_{ε}(z, a) = max(0, min(a, z)) ⋆ (G_{ε}(z)G_{ε}(a)).

The starting wave can be defined by:

φ_{ε}(t, x) = x + f_{ε}(t − λx, g(x)), λ < 1, g increasing.

Note that the path of an individual particle x hits the wave at t = λx − ε and leaves it at t = λx + g(x) + ε, having moved forward to x + g(x). Calculate the derivatives:

(f_{ε})_{z} = I_{0≤z≤a} ⋆ (G_{ε}(z)G_{ε}(a)) ∈ [0, 1]
(f_{ε})_{a} = I_{0≤a≤z} ⋆ (G_{ε}(z)G_{ε}(a)) ∈ [0, 1]
(φ_{ε})_{t} = (f_{ε})_{z}(t − λx, g(x))
(φ_{ε})_{x} = 1 − λ(f_{ε})_{z}(t − λx, g(x)) + (f_{ε})_{a}(t − λx, g(x)) · g′(x) > 0.

This gives us:

E_{t_{0}}^{t_{1}}(φ) = ∫_{t_{0}}^{t_{1}} ∫_{ℝ}(φ_{ε})_{t}^{2}(φ_{ε})_{x}dx dt
≤ ∫_{t_{0}}^{t_{1}} ∫_{ℝ}(f_{ε})_{z}^{2}(t − λx, g(x)) · (1 − λ(f_{ε})_{z}(t − λx, g(x)))dx dt
+ ∫_{t_{0}}^{t_{1}} ∫_{ℝ}(f_{ε})_{z}^{2}(t − λx, g(x)) · (f_{ε})_{a}(t − λx, g(x))g′(x)dx dt

The first integral can be bounded as in the original discussion. The second integral is also small because the support of the z-derivative is −ε ≤ t − λx ≤ g(x) + ε, while the support of the a-derivative is −ε ≤ g(x) ≤ t − λx + ε, so together |g(x) − (t − λx)| ≤ ε. Now define x_{1} and x_{2} by g(x_{1}) + λx_{1} = t + ε and g(x_{0}) + λx_{0} = t − ε. Then the inner integral is bounded by

∫_{|g(x)+λx−t|≤ε} g′(x)dx = g(x_{1}) − g(x_{0}) ≤ 2ε,

and the whole second term is bounded by 2ε(t_{1} − t_{0}). Thus the length is O(ε).

The end of the wave can be handled by playing the beginning backwards. If the distance that a point x moves when the wave passes it is to be g(x), so that the final diffeomorphism is x ↦ x + g(x) then let b = max(g) and use the above definition of φ while g′ > 0. The modification when g′ < 0 (but g′ > −1 in order for x ↦ x + g(x) to have positive derivative) is given by:

φ_{ε}(t, x) = x + f_{ε}(t − λx − (1 − λ)(b − g(x)), g(x)).

A figure showing the trajectories φ_{ε}(t, x) for sample values of x is shown in the figure above.

It remains to show that Diff_{0}(N)^{E=0} is a nontrivial subgroup for an arbitrary Riemannian manifold. We choose a piece of a unit speed geodesic containing no conjugate points in N and Fermi coordinates along this geodesic; so we can assume that we are in an open set in ℝ^{m} which is a tube around a piece of the u^{1}-axis. Now we use a small bump function in the the slice orthogonal to the u^{1}-axis and multiply it with the construction from above for the coordinate u^{1}. Then it follows that we get a nontrivial diffeomorphism in Diff_{0}(N)^{E=0} again. □

Remark. Theorem 5.2 can possibly be proved directly without the help of the simplicity of Diff_{0}(N). For N = ℝ one can use the method of 5.2, (7) in the parameter space of a curve, and for general N one can use a Morse function on N to produce a special coordinate for applying the same method, as we did in the proof of theorem 3.2.

5.3. Geodesics and sectional curvature for G^{0} on Diff(N). According to Arnold [1], see [11], 3.3, for a right invariant weak Riemannian metric G on an (possibly infinite dimensional) Lie group the geodesic equation and the curvature are given in terms of the adjoint operator (with respect to G, if it exists) of the Lie bracket by the following formulas:

u_{t} = − ad(u)^{∗}u, u = φ_{t} ∘ φ^{−1}
G(ad(X)^{∗}Y, Z) := G(Y, ad(X)Z)
4G(R(X, Y)X, Y) = 3G(ad(X)Y, ad(X)Y) − 2G(ad(Y)^{∗}X, ad(X)Y)

− 2G(ad(X)^{∗}Y, ad(Y)X) + 4G(ad(X)^{∗}X, ad(Y)^{∗}Y)

− G(ad(X)^{∗}Y + ad(Y)^{∗}X, ad(X)^{∗}Y + ad(Y)^{∗}X)

In our case, for Diff_{0}(N), we have ad(X)Y = −[X, Y] (the bracket on the Lie algebra 𝔛_{c}(N) of vector fields with compact support is the negative of the usual one), and:

G^{0}(X, Y) = ∫_{N} g(X, Y) vol(g)
G^{0}(ad(Y)^{∗}X, Z) = G^{0}(X, −[Y, Z]) = ∫_{N} g(X, −L_{Y} Z) vol(g)

= ∫_{N} g(L_{Y} X + (g^{−1}L_{Y} g)X + div^{g}(Y)X, Z)vol(g)
ad(Y)^{∗} = L_{Y} + g^{−1}L_{Y} (g) + div^{g}(Y) Id_{T} N = L_{Y} + β(Y),

where the tensor field β(Y) = g^{−1}L_{Y} (g)+div^{g}(Y) Id : T N → T N is self adjoint with respect to g. Thus the geodesic equation is

u_{t} = −(g^{−1}L_{u}(g))(u) − div^{g}(u)u = −β(u)u, u = φ_{t} ∘ φ^{−1}.

The main part of the sectional curvature is given by:

4G(R(X, Y)X, Y) =

= ∫_{N}(3∥[X, Y]∥_{g}^{2} + 2g((L_{Y} + β(Y))X, [X, Y]) + 2g((L_{X} + β(X))Y, [Y, X])

+ 4g(β(X)X, β(Y)Y) − ∥β(X)Y + β(Y)X∥_{g}^{2}) vol(g)

= ∫_{N}(−∥β(X)Y − β(Y)X + [X, Y]∥_{g}^{2} − 4g([β(X), β(Y)]X, Y)) vol(g)

So sectional curvature consists of a part which is visibly non-negative, and another part which is difficult to decompose further.

5.4. Example: Burgers’ equation. For (N, g) = (ℝ, can) or (S^{1}, can) the geodesic equation is Burgers’ equation [2], a completely integrable infinite dimensional system,

u_{t} = −3u_{x} u, u = φ_{t} ∘ φ^{−1}

and we get G^{0}(R(X, Y)X, Y) = − ∫ [X, Y]^{2} dx so that all sectional curvatures are non-negative.

5.5. Example: n-dimensional analog of Burgers’ equation. For (N, g) = (ℝ^{n}, can) or ((S^{1})^{n}, can) we have:

(ad(X)Y)^{k} = ∑_{i}((∂_{i}X^{k})Y^{i} − X^{i}(∂_{i}Y^{k}))
(ad(X)^{∗}Z)^{k} = ∑_{i}((∂_{k}X^{i})Z^{i} + (∂_{i}X^{i})Z^{k} + X^{i}(∂_{i}Z^{k})),

so that the geodesic equation is given by

∂_{t}u^{k} = −(ad(u)^{⊤}u)^{k} = − ∑_{i}((∂_{k}u^{i})u^{i} + (∂_{i}u^{i})u^{k} + u^{i}(∂_{i}u^{k})),

called the basic Euler-Poincaré equation (EPDiff) in [6], the n-dimensional analog of Burgers’ equation.

5.6. Stronger metrics on Diff_{0}(N). A very small strengthening of the weak Riemannian H^{0}-metric on Diff_{0}(N) makes it into a true metric. We define the stronger right invariant semi-Riemannian metric by the formula:

G_{φ}^{A}(h, k) = ∫_{N}(g(X, Y) + A div_{g}(X). div_{g}(Y)) vol(g).

Then the following holds:

5.7. Theorem. For any distinct diffeomorphisms φ_{0}, φ_{1}, the infimum of the lengths of all paths from φ_{0} to φ_{1} with respect to G^{A} is positive.

This implies that the metric G^{0} induces positive geodesic distance on the subgroup of volume preserving diffeomorphism since it coincides there with the metric G^{A}.

Proof. Let ψ_{1} = φ_{0} ∘ φ_{1}^{−1}. If φ_{0} ≠ φ_{1}, there are two functions ρ and f on N with compact support such that:

∫_{N} ρ(y)f(ψ_{1}(y)) vol(g)(y) ≠ ∫_{N} ρ(y)f(y) vol(g)(y).

Now consider any path φ(t, y) between the two maps with derivative u = φ_{t} ∘ φ^{−1}. Inverting the diffeomorphisms (or switching from a Lagrangian to an Eulerian point of view), let ψ(t, ) = φ(0, )∘φ(t, )^{−1}. Then ψ_{t} = −T ψ(u) and we have:

∫Nρ(y)f(ψ_{1}(y)) vol(g)(y) − ∫_{N} ρ(y)f(y) vol(g)(y) =
= ∫_{0}^{1} ∫_{N} ρ(y)∂tf(ψ(t, y) vol(g)(y)dt
= ∫_{0}^{1} ∫_{N} ρ(y)(df ∘ ψ)(ψ_{t}(t, y)) vol(g)(y) dt
= ∫_{0}^{1} ∫_{N} ρ(y)(T f ∘ ψ)(−T ψ(u(t, y))) vol(g)(y)dt

But div((f ∘ ψ) · ρu) = (f ∘ ψ) · div(ρu) + (T f ∘ ψ)(T ψ(ρu)). The integral of the left hand side is 0, hence:

| ∫_{N} ρ(y)f(ψ_{1}(y)) vol(g)(y) − ∫_{N} ρ(y)f(y) vol(g)(y)|
= | ∫_{0}^{1} ∫_{N}(f ∘ ψ) div(ρu) vol(g)dt|

≤ sup(|f|) ∫_{0}^{1} \sqrt{∫_{N} C_{ρ}∥u∥^{2} + C′_{ρ}| div(u)|^{2} vol(g)}dt

for constants C_{ρ}, C′_{ρ} depending only on ρ. Clearly the right hand side is a lower bound for the length of any path from φ_{0} to φ_{1}. □

5.8. Geodesics for G^{A} on Diff(ℝ). See [3] and [12]. We consider the groups Diff_{c}(ℝ) or Diff(S^{1}) with Lie algebras 𝔛_{c}(ℝ) or 𝔛(S^{1}) with Lie bracket ad(X)Y = −[X, Y] = X′Y − XY′. The G^{A}-metric equals the H^{1}-metric on 𝔛_{c}(ℝ), and we have:

G^{A}(X, Y) = ∫_{ℝ}(XY + AX′Y′)dx = ∫_{ℝ} X(1 − A∂_{x}^{2})Y dx,
G^{A}(ad(X)^{∗}Y, Z) = ∫_{ℝ}(Y X′Z − Y XZ′ + AY′(X′Z − XZ′)′)dx

= ∫_{ℝ} Z(1 − ∂_{x}^{2})(1 − ∂_{x}^{2})^{−1}(2Y X′ + Y′X − 2AY′′X′ − AY′′′X)dx,

ad(X)^{∗}Y = (1 − ∂_{x}^{2})^{−1}(2Y X′ + Y′X − 2AY′′X′ − AY′′′X)
ad(X)^{∗} = (1 − ∂_{x}^{2})^{−1}(2X′ + X∂_{x})(1 − A∂_{x}^{2})

so that the geodesic equation in Eulerian representation u = (∂_{t}f)∘f^{−1} ∈ 𝔛_{c}(ℝ) or 𝔛(S^{1}) is

∂_{t}u = − ad(u)^{∗}u = −(1 − ∂_{x}^{2})^{−1}(3uu′ − 2Au′′u′ − Au′′′u), or
u_{t} − u_{txx} = Au_{xxx}.u + 2Au_{xx}.u_{x} − 3u_{x}.u,

which for A = 1 is the Camassa-Holm equation [3], another completely integrable infinite dimensional Hamiltonian system. Note that here geodesic distance is a well defined metric describing the topology.

References

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[2] J. Burgers. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1 (1948), 171–199.

[3] R. Camassa and D. Holm. An integrable shallow water equation with peaked solutions. Phys. Rev. Lett. 71 (1993), 1661-1664.

[4] Vicente Cervera, Francisca Mascaro, and Peter W. Michor. The orbit structure of the action of the diffeomorphism group on the space of immersions. Diff. Geom. Appl., 1 (1991), 391–401.

[5] D.B.A. Epstein. The simplicity of certain groups of homeomorphisms. Compositio Math. 22 (1970), 165–173.

[6] D. Holm, T. Ratnanather, A. Trouve and L. Younes. Soliton Dynamics in Computational Anatomy NeuroImage, Sept. 2004.

[7] Gerd Kainz. A metric on the manifold of immersions and its Riemannian curvature. Monatsh. Math., 98 (1984), 211-217.

[8] Andreas Kriegl and Peter W. Michor. The Convenient Setting for Global Analysis. AMS, Providence, 1997. ‘Surveys and Monographs 53’.

[9] John N. Mather. Commutators of diffeomorphisms I, II. Comm. Math. Helv. 49 (1974), 512–528; 50 (1975), 33–40.

[10] Peter W. Michor and David Mumford. Riemannian geometries on spaces of plane curves. arXiv:math.DG/0312384.

[11] Peter W. Michor and Tudor Ratiu. Geometry of the Virasoro-Bott group. J. Lie Theory, 8 (1998), 293–309. arXiv:math.DG/9801115

[12] G. Misiołek. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24 (1998), 203–208.

[13] Richard Palais. On the existence of slices for actions of non-compact Lie groups. Ann. of Math. 73 (1961), 295–323.

[14] William Thurston. Foliations and groups of diffeomorphisms. Bull. AMS 80 (1974), 304–307.

Peter W. Michor Fakultät für Mathematik Universität Wien Nordbergstrasse 15 A-1090 Wien, Austria and Erwin Schrödinger Institut für Mathematische Physik Boltzmanngasse 9 A-1090 Wien, Austria Peter.Michor@univie.ac.at David Mumford Division of Applied Mathematics Brown University Box F, Providence, RI 02912, USA David Mumford@brown.edu 246

Documenta Mathematica 10 (2005)