Definition of manifolds of sections of a Riemannian sub-

II.2.3 Definition of manifolds of sections of a Riemannian

u∈Γ^k(W) and consider the Riemannian vector bundle u^∗V W → Σ. In addi-tion, assume that u has bounded derivatives up to order k, i. e. C^u,r <∞ for all r = 1, . . . , k. There is then a Sobolev space L^k,p(u^∗V W, h, g,∇^⊥) that is defined as a Banach space in one of the usual ways (completion of the space of smooth section or via weak derivatives). The important properties these have to satisfy are the following, irrespective of the precise definition:

First of all, the smooth sections of finite L^k,p-norm form a dense linear sub-space, cf. Proposition 3.2, p. 15, in [Eic07].

Second, the Sobolev embedding theorems hold, cf. Theorem 3.4, p. 16, in [Eic07].

And third, the module structure theorem holds, cf. Theorem 3.12, p. 20, in [Eic07].

Apart from the fact that the boundedness assumption on the covariant deriva-tives of D^vu are necessary for the embedding and module structure theorems quoted above to hold, they are also necessary to be able to use Proposition II.4 in the construction of the transition functions later on.

But on the other hand, these are the only results about the (linear) Sobolev space needed for the construction below, so one can easily replace them by weighted Sobolev spaces, for example.

To shorten notation, in the following, L^k,p(u^∗V W) will be written instead of L^k,p(u^∗V W, h, g,∇^⊥).

Since kp > n := dim Σ, the Sobolev embedding theorem shows the existence of a canonical continuous injection Ψ_u : L^k,p(u^∗V W) ,→ Γ⁰(u^∗V W). As a consequence, the map Φ_u : L^k,p(u^∗V W) → Γ⁰(W), ξ 7→ (z 7→ exp^⊥_u(z)(ξ(z))), is well defined and continuous as it factors through Ψu. The next step is to find a neighbourhoodV_u of the zero section in L^k,p(u^∗V W) that gets mapped injectively by Φ_uinto Γ⁰(W), to serve as a chart of a Banach manifold structure.

Here a problem arises if Σ is noncompact. Ideally, injectivity should result from pointwise injectivity, which requires that for every z ∈ Σ, kΨ_u(ξ)(z)k ≤ inj_Wz, where inj_Wz denotes the injectivity radius of the fibreW_z over the point z ∈ Σ. Because the existence of a Sobolev constant s^Σ,k,p was assumed and remembering that the constant δ from Definition II.13 bounds the injectivity radii of theW_z from below, since the W_z are complete, one can take the open subset Vu := {ξ ∈ L^k,p(u^∗V W) | kξk_Lk,p < δ/(3s^Σ,k,p)}. Anyway, one can tentatively define

L^k,p(W, π, g) := [

u∈Γ^k(W) C^u,r<∞, r=1,...,k

Φ_u(V_u)⊆Γ⁰(W).

The topology on this set is defined to be the topology generated by the union of the induced topology onL^k,p(W, π, g) as a subset of Γ⁰(W) and the topologies on the subsets Φ_u(V_u) induced by the Banach space topologies on the V_u ⊆ L^k,p(u^∗V W). This is clearly a Hausdorff space, the topology being finer than the (Hausdorff) topology on Γ⁰(W). But it is not yet clear that it is 2^nd -countable. At this point a caveat has to be issued, for it is not clear that

the natural candidates for coordinate maps, the Φ_u, are homeomorphisms onto their image, with continuity possibly failing (they are continuous w. r. t. the C⁰-topology, but this does not imply continuity w. r. t. the finer topology on L^k,p(W, π, g)), although they are open injections by definition of the topology on L^k,p(W, π, g). Another way to phrase this is by noting that the natural candidate for a transition function, Φ⁻¹_u ◦Φv for Φu(Vu)∩Φv(Vv)6=∅, restricted to Φ⁻¹_v (Φ_u(V_u))⊆V_v, need not be a homeomorphism between open subsets of Vv andVu.

So assume that there exist ζ ∈ Vv, ξ ∈ Vu with Φv(ζ) = Φu(ξ). As before, denote Φu(ξ) =: u_ξ = v_ζ := Φ_v(ζ) ∈ Γ⁰(W). Then for all ξ⁰ ∈ Vu and z ∈ Σ, d(v(z),Φ_u(ξ⁰)(z)) ≤ d(v(z), v_ζ(z)) +d(u_ξ(z), u(z)) +d(u(z),Φ_u(ξ⁰)(z)) < δ by definition of Vu. And analogously for ζ⁰ ∈ Vv with the roles of u and v exchanged. This implies that there are well-defined maps, by abuse of notation,

Φ⁻¹_u ◦Φ_v:V_v →Γ⁰(u^∗V W) and

Φ⁻¹_v ◦Φ_u :V_u →Γ⁰(v^∗V W).

Denote V_v^k := Vv∩Γ^k(v^∗V W) and analogously V_u^k := Vu∩Γ^k(u^∗V W). Then forη∈L^k,p(v^∗V W)∩Γ^k(v^∗V W) small enough, denoteζ⁰=ζ⁰(η) :=ζ+η∈V_v^k. Then from Proposition II.4 follow pointwise estimates

∂

∂η∇^mΦ⁻¹_u ◦Φv(ζ⁰)

p

≤C

m

X

k⁰=0

X

r,i,k⁰_j r

Y

s=1

k∇^k0sζ⁰k^pk∇ⁱηk^p,

whereCis a constant that depends on all the bounds above, esp. C^u,r andC^v,r forr= 1, . . . , k, but not on anything else. Integration over Σ and applying the module structure theorem, form= 0, . . . , k, then gives a global estimate

∂

∂ηΦ⁻¹_u ◦Φv(ζ⁰) L^k,p

≤C⁰kζ⁰k_Lk,pkηk_Lk,p.

The mean value theorem then implies that forζ⁰∈V_u^k, Φ⁻¹_u ◦Φ_v(ζ⁰)∈L^k,p(u^∗V W).

In particular, for η small enough, i. e. ζ⁰ close enough to ζ, in L^k,p-norm, Φ⁻¹_u ◦Φv(ζ⁰) ∈ V_u^k. The same clearly also holds with the roles of u and v interchanged. This implies that

Φ⁻¹_u ◦Φ_v : Φ⁻¹_v (Φ_u(V_u^k))→V_u^k

is a well-defined Lipschitz continuous map that hence has a well-defined Lips-chitz continuous completion to a map

Φ⁻¹_u ◦Φv : Φ⁻¹_v (Φu(Vu))→Vu.

In particular, Φ⁻¹_v (Φu(Vu))⊆Vv is an open subset. Again, the same holds with the roles of u and v interchanged and the resulting maps are inverses to each

other.

Now the same line of arguments above using Proposition II.4 and the module structure theorem implies that the map

Φ⁻¹_u ◦Φ_v : Φ⁻¹_v (Φ_u(V_u^k))→V_u^k

has Lipschitz continuous derivatives of all orders. By Corollary A.1, this in turn implies that the completion

Φ⁻¹_u ◦Φ_v : Φ⁻¹_v (Φ_u(V_u))→V_u. is smooth.

So far, this turnsL^k,p(W, π, g) into an a priori non-2^nd-countable Banach man-ifold. If Σ is compact, then the standard argument via choosing a C^k-dense countable subset {u_i}_i∈N of Γ^k(W) and in the charts above around each ui

a L^k,p-dense countable subset applies, for the boundedness condition on the covariant derivatives of the D^vu_i are automatically satisfied and any smooth section ofu^∗V W lies automatically inL^k,p(u^∗V W).

In general, for the above to carry over, one immediate condition is that the Banach spaces L^k,p(u^∗V W) need to be separable. But if u, u⁰ ∈Γ^k(W) satisfy C^u,r, C^u0,r <∞forr= 1, . . . , r, thenuandu⁰ C^k-close does not imply that one lies in the chart around the other. So it does not suffice to take dense subsets of theV_u foruin a C^k-dense subset of Γ^k(W) as before.

In practice, it is then easier to just restrict the set ofC^k-sections around which L^k,p(W, π, g) is constructed to a countable subset, tailored to the concrete prob-lem.

An example for this would be the maps with cylindrical ends used in SFT-Fredholm theory.

In the compact case, i. e.W and hence Σ compact, all of these problems vanish, because all the assumptions on finiteness of the constantsC^∗,r are automatically satisfied. Also in this case the constants C^g,˜g,r appearing in Proposition II.4, in case π : W → Σ is equipped with two different structures of Riemannian submersion, are automatically finite, which implies that the Banach manifold structure on L^k,p(W, π, g) is independent of g.

Lemma II.10. If π:W →Σis a Riemannian submersion andW is compact, thenL^k,p(W, π, g)is a2^nd-countable Hausdorff Banach manifold and the under-lying set as well as the Banach manifold structure on this set are independent of g.

Proof. Only independence of the Riemannian structure needs to be shown.

For this, what one wants to show is that the set-theoretic identity defines a smooth map between the Banach manifolds built with respect to two different choices of Riemannian structures, given by metricsg and ˜g on W. Expressing the identity in local charts around a point u ∈Γ^k(W) means that one has to look at maps of the form ˜Φ⁻¹_u ◦Φ_u, where ˜Φ_u is as in the construction above,

but w. r. t. the metric ˜g. The proof that this defines a smooth map between open subsets then proceeds literally as the corresponding proof outlined in the construction above using Proposition II.4.

One last easy consequence of the construction ofL^k,p(W, π, g) above is the non-linear version of the Sobolev embedding theorem. For this, note that one can construct Banach manifolds of sections of classC^kvery analogously to theL^k,p -spaces above. Given a vector bundle ρ : E → B together with a Riemannian metric h on B, a fibre metricg on E and a metric connection ∇, the space of sections of class C^k, for k∈N0, is the Banach space (k · k∞ denotes the usual supremums-norm on functions)

Γ^k(E, h, g,∇) :={ξ ∈Γ^k(E)|

k

X

i=0

k|∇ⁱξ|k_∞<∞},

where |∇ⁱξ| ∈C^k(B) denotes the norm on (ΛⁱT^∗B)⊗E induced byh and g.

Construction II.3. If π : W → Σ is a Riemannian submersion of bounded geometry, then in the notation of Definition II.13, for u ∈Γ^k(W) withC^u,r <

∞ ∀r= 1, . . . , k, let

Uu :={ξ∈Γ^k(u^∗V W, h, g,∇^⊥)| k|ξ|k_∞< δ}.

Then analogously to the previous construction, there are well-defined injective maps

Ψ_u:U_u→Γ^k(W).

Using these, one defines the space of sections of class C^k of π:W →Σ as Γ^k(W, π, g) := [

u∈Γ^k(W) C^u,r<∞, r=1,...,k

Φu(Vu)⊆Γ⁰(W).

It follows directly from Proposition II.4 that the maps Ψu define an atlas for a Banach manifold structure on Γ^k(W, π, g).

Then the following nonlinear version of the Sobolev embedding theorem is an immediate consequence of the linear Sobolev embedding theorem and the con-structions of the Banach manifolds involved.

Lemma II.11. Let π:W →Σbe a Riemannian submersion of bounded geom-etry. Let furthermore k, ` ∈N0 an p ∈(1,∞) with k−<sup>dim Σ</sup><sub>p</sub> > `. Then there is a smooth embedding

L^k,p(W, π, g),→Γ^`(W, π, g),

defined by the restriction of the set-theoretic identity on Γ⁰(W).

Im Dokument Universal moduli spaces in Gromov-Witten theory (Seite 56-61)