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Pinching on open manifolds

We show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold carries arbitrarily pinched (incomplete) Riemannian metrics.


Introduction
A basic question in Riemannian geometry is what effect the existence of a Riemannian metric with particular curvature properties has on the topology of the underlying manifold. One usually considers complete metrics and obtains even on an elementary level rather strong restrictions. For instance, complete manifolds of negative sectional curvature are always aspherical and all nontrivial elements of the fundamental group have infinite order. Furthermore a closed manifold does not support two metrics of different signed sectional curvatures. We also mention the result of Gromoll and Meyer [3] that a complete open manifold of positive sectional curvature is diffeomorphic to some R m .
The situation changes completely if we skip the completeness assumption, which is automatically imposed on metrics on closed manifolds but is an extra condition on open manifolds. Gromov remarks in his thesis [4] the astonishing fact that every open manifold carries metrics with strict positive and negative sectional curvature.
In this paper we extend this result and prove the existence of arbitrarily pinched (incomplete) metrics on open manifolds. In other words the geometric significance of the sign of the curvature and even of curvature bounds depends on the completeness of the underlying metric. It turns out that the only question concerning curvature which involves geometry in the incomplete case is the existence of metrics of constant sectional curvature, since then topological obstructions are known. For example such manifolds have trivial Pontryagin classes (cf. Theorem 43 and Corollary 44 in [9]). Theorem 1.1 is an application of a deep differential topological insight of M. Gromov who greatly generalized in his thesis [4] Smale-Hirsch-Phillips immersion-submersion theory (see [8], [6], [7]) by proving what is now called the h-principle for invariant, open differential relations on open manifolds.
Roughly speaking, a partial differential relation R is any condition imposed on the partial derivatives of an unknown function. By substituting derivatives by new independent variables one gets an underlying algebraic relation. Obviously the existence of a formal solution, i. e. a solution of the corresponding algebraic relation, is a necessary condition for the solvability of R. It turns out that under certain cirumstances any formal solution of R can be deformed into a genuine one.
In general one considers a smooth fibre bundle over the underlying manifold. A partial differential relation is a condition imposed on the r−jet bundle of local sections. Now we could try and construct a section of this r−jet bundle which pointwise fulfills the relation, i.e. a formal solution, and deform it to get a real solution. In other words, we reduce the problem to algebraic-topological obstruction theory. We refer to [2] and [5] for expositions of this technique.
In our case we study the bundle of symmetric bilinear forms the positive definite sections of which are Riemannian metrics on the underlying manifold. The observation that the sectional curvature only depends on the 2−jet of the metric allows us to translate the pinching problem into a curvature relation, see Definition 3.2. In Lemma 3.5 we will prove that on any manifold there are formal solutions with constant curvature. Then, as an application of Gromov's h-principle, Theorem 3.4 implies the existence of arbitrarily pinched metrics.

Some basic facts concerning jets
Given a (C ∞ -)smooth fibre bundle q : V → M over a smooth manifold M of dimension m we identify two local smooth sections σ 1 and σ 2 defined in a neighbourhood of some point p ∈ M if in local local coordinates on M and V they have the same partial derivatives up to order r at p. An equivalence class [(σ, p)] = j r p σ under this relation is called r-jet of σ at p. We denote by V r the space of r-jets of local sections. A partial differential relation of order r is a subset R ⊂ V r .
In the following we consider the bundle q : We consider the projection map Let p ∈ M and j 2 p g ∈ E 2 a 2-jet represented by a local Riemannian metric g. Using a chart (U, Φ, x 1 , ..., x m ) around p, we obtain a local description of j 2 p g taking symmetries of the partial derivatives into account. Vice versa, . As a consequence there is a 1-1 correspondence Now let (W, Ψ, y 1 , ..., y m ) be another chart defined near p. Keeping in mind that

The curvature relation
Let τ = j 2 p g be a 2-jet of a local Riemannian metric g at p ∈ M and (U, Φ, x 1 , ..., x m ) a chart around p. We may assume that g is defined on U. In local coordinates τ is associated to ). On U the metric g induces the Levi-Civita connection ∇ given locally by means of the Christoffelsymbols defined by . We note that the entries (g Φ ) ij are rational functions in terms of the g Φ ij . It follows that the components R ijks of the curvature tensor given by ∂ ∂x s at p are completely determined by τ and independent of the choice of the representative g. This implies that the sectional curvature of a plane V ⊂ T p M spanned by two linearly independent vectors v, w ∈ T p M only depends on τ. In other words, we have a well-defined notion of sectional curvature of 2-jets of local Riemannian metrics.
Definiton 3.1. Let τ ∈ E 2 be a 2-jet at p ∈ M of a Riemannian metric g on a neighbourhood U of p and V a plane in T p M. We define the sectional curvature sec τ V as the sectional curvature sec (U,g) V with respect to the Levi-Civita connection induced by g on U.
Definiton 3.2. Given a ∈ R and δ ≥ 0 the curvature relation R δ,a is defined as the subset R δ,a = {τ ∈ E 2 : sec τ V ∈ (a − δ, a + δ) for all planes V ⊂ T q 2 (τ ) M } where we think of (a, a) as {a} if δ = 0 by abuse of notation.
We call a continuous section α of E 2 satisfying α(M ) ⊂ R δ,a a formal solution to the curvature relation R δ,a . A formal solution α is holonomic if j 2 g = α for some section g of E. In other words, g is a Riemannian metric on M such that all sectional curvatures lie in (a − δ, a + δ). We refer to [2] and [5] for these notions in the broader context of partial differential relations.
Let M 1 and M 2 be two differentiable manifolds of the same dimension. As above we define bundles q i : E i → M i and q 2 i : E 2 i → M i as well as curvature relations R i δ,a with i ∈ {1, 2}. A local diffeomorphism f : U 1 → U 2 between open subsets U 1 ⊂ M 1 and U 2 ⊂ M 2 induces in a canonical way a map f * : E 2 1 |U 1 → E 2 2 |U 2 as follows: Let τ = j 2 p g ∈ E 2 1 . We may assume that its representative g is defined on U 1 . The push-forward f * g yields a metric on U 2 via for all v, w ∈ TpM 2 and arbitraryp ∈ U 2 . Now we define One readily checks that f * τ is well-defined. Suppose that τ = j 2 p g ∈ R 1 δ,a . Due to the fact that f : (U 1 , g) → (U 2 , f * g) is an isometry we have where E(v, w) is the plane spanned by two linearly independent vectors v, w ∈ T f (p) U 2 . In other words, f * τ ∈ R 2 δ,a . In this sense curvature relations are invariant under local diffeomorphisms. Proof. Let p ∈ M and (U, Φ) a chart of M near p. We obtain a local trivialisation (q 2 ) −1 (U ) ∼ = U × P (m) 2 of the bundle q 2 : E 2 → M. Any 2-jet τ ∈ R δ,a has a local representation We note that the sectional curvature depends only on (τ ij , τ ijk , τ ijkl ). Consequently,τ = (p, τ ij , τ ijk , τ ijkl ) ∈ R δ,a for anyp ∈ U. We define F δ,a as a subset of P (m) 2 in the following way: (τ ij , τ ijk , τ ijkl ) ∈ F δ,a if and only if for some (and hence any)p ∈ U we havẽ In other words, (q 2 ) −1 (U ) ∩ R δ,a ∼ = U × F δ,a . As a result, q 2 : R δ,a → M is trivial over charts of M. A change of coodinates on M induces a linear transformation of F δ,a ⊂ P (m) 2 , because the sectional curvature is independent of the choice of local coordinates. Now assume δ > 0. We identify T p M with R m and parametrize the set of all planes in T p M by the Stiefel manifold V m,2 of the orthonormal 2-frames (v, w) ∈ R m . Furthermore, let E(v, w) be the plane in R m spanned by two linearly independent vectors v and w. The function is continuous and V m,2 is compact. It follows that F δ,a is an open subset of P (m) 2 .
Let δ > 0. We denote by ΓR δ,a the space of formal solutions equipped with the compact-open topology and write for the space of smooth sections of E, i.e. Riemannian metrics on M, such that all sectional curvatures lie in (a−δ, a+δ). The map j 2 : Γ R δ,a E → ΓR δ,a induces the weak C 2 −topology on Γ R δ,a E.
is a weak homotopy equivalence.
In other words, if we can prove that there exist formal solutions we are done. Surjectivity on π 0 then yields arbitrarily pinched Riemannian metrics.
Lemma 3.5. The space ΓR δ,a of formal solutions is nonempty for any a ∈ R and δ ≥ 0.
Proof. It suffices to show that R 0,a ⊂ R δ,a admits formal solutions. We will prove that the fibre F 0,a =: F a smoothly deformation retracts to an arbitrary element of F a .
Note that F a is nonempty. In a neighbourhood of 0 ∈ R m we choose a Riemannian metric g with constant sectional curvature a. Then Let τ be the Taylor polynom which represents the 2-jet (0, τ ′ ) ∈ R m × F a . We regard τ as a Riemannian metric defined in a neighbourhood of 0 ∈ R m . Now we choose a reference basis B = {b 1 , ..., b m } of R m . Applying the Gram-Schmidt procedure yields an orthonormal basis B τ w.r.t. τ (0). We change to normal coordinates centered at 0 w.r.t. τ and B τ , i.e. we identify T 0 R m with R m via B τ , and the exponential map induces a linear and invertible transformatioñ where δ ij = 1 if i = j and δ ij = 0 if i = j. Let ψ ′ ∈ P (m) 2 and ψ the Taylor polynom associated to (0, ψ ′ ) ∈ R m × P (m) 2 . Like τ we think of ψ as a Riemannian metric defined near 0 ∈ R m .
is a Riemannian metric in a neighbourhood of 0 ∈ R m with ψ 0 = ψ und ψ 1 = τ. We write with respect to ψ t (0) such that the maps We claim that L(t, ψ ′ ) depends smoothly on (t, ψ ′ ) (and so does L(t, ψ ′ ) −1 by Kramer's rule).
Thus we conclude h(s, ψ ′ ) ∈ F a for all s ∈ [0, 1]. Taking into account that the transformations L(t, ψ ′ ) are induced by coordinate changes we have G(t, ψ ′ ) = L(t, ψ ′ ) −1 (h(t, ψ ′ )) ∈ F a for all t ∈ [0, 1]. It follows that F a is contractible and by elementary obstruction theory there exists a global section of q 2 : R 0,a → M, i.e. a formal solution.