Thomas Schick

Manifolds with boundary and of bounded geometry

last compiled: October 21, 1998— last edited: after October 17, 1998

Abstract. For (non-compact) manifolds with boundary we prove that bounded geometry defined by global curvature bounds is equivalent to bounded geometry defined using bounds on geodesic charts.

We produce a nice atlas with subordinate partition of unity on boundary-manifolds of bounded geometry, and we study the change of (geodesic) coordinate maps.

1. Introduction

Manifolds of bounded geometry arise naturally when one deals with noncompact Riemannian manifolds, and are studied extensively in the literature. So far, the focus was on manifolds without boundary.

One main source of examples are coverings of compact manifolds, which are particularly important in the context of L^{2}-cohomology and other L^{2}- invariants. These are studied frequently also for manifolds with boundary. Therefore, it is natural to look at more general manifolds with boundary and bounded geometry.

There are mainly two ways to define manifolds of bounded geometry: either one uses bounds on the curvature (and its covariant derivatives) —this is the global description— or one uses geodesic charts and bounds on the metric tensor and its derivatives in these coordinates —the local approach. A proof of the equivalence of these two approaches for manifolds without boundary can be found in Eichhorn [3].

Dealing with manifolds with boundary, in addition to the usual requirements in the interior we must impose boundary regularity conditions. These involve the second fundamental form (in the global description) or special charts for the boundary (in the local description).

The main aim of this paper is to give the two definitions here and to prove that they are in fact equivalent.

In the last section, we show that the change of (geodesic) coordinate functions and their derivatives admit uniform bounds on manifolds of bounded

This work was finished during a stay at Penn State University funded by the DAAD FB Mathematik, Universität Münster; and Dept. of Mathematics, Penn State University email: thomas.schick@math.psu.edu

geometry. And we provide one technical tool, namely a nice atlas with subordinate nice partition of unity. This was introduced and used by Shubin [8, 1.2 and 1.3] if the boundary is empty.

This paper is part of the Dissertation [6] of the author, and the results obtained here are used in [7]. I thank my advisor, Prof. Wolfgang Lück, for his constant support and encouragement.

2. Global versus local curvature bounds

We start with the (global) definition of manifolds of bounded geometry.

Definition 1. Suppose M is a manifold with boundary ∂M (possibly empty). It is of (global) bounded geometry if the following holds:

(N) Normal collar: there exists r_{C} > 0 so that the geodesic collar

N = [0, r_{C}) × ∂M → M : (t, x) ↦ exp_{x}(tν_{x})

is a diffeomorphism onto its image. Here ν_{x} is the unit inward normal vector at x ∈ ∂M. Equip N with the induced Riemannian metric. In the sequel we will identify N and its image. Set N_{1/3} := im[0, r_{C}/3)×∂M and define N_{2/3} similarly.

(IC) The injectivity radius of ∂M is positive: r_{inj}(∂M) > 0. (I) Injectivity radius of M: There is r_{i} > 0 so that for x ∈ M − N_{1/3}

the exponential map is a diffeomorphism on B(0, r_{i}) ⊂ T_{x}M. I.e. if we identify T_{x}M with ℝ^{m} via an orthonormal frame we have Gaussian coordinates ℝ^{m} ⊃ B(0, r_{i}) →^{κ_{x}} M around every point in M − N_{1/3}.

(B) Curvature bounds: For every K ∈ ℕ there is C_{K} > 0 so that |∇^{i}R| ≤ C_{K} and | ¯∇^{i}l| ≤ C_{K} ∀0 ≤ i ≤ K. Here R is the curvature and l the second fundamental form tensor (with respect to ν), ∇ is the Levi-Civita connection of M and ¯∇ the one of ∂M

The injectivity radius and curvature bounds are what one is used to for manifolds without boundary [2, section 3]. The embedding of the boundary is described by the second fundamental form. Because the injectivity radius does not make sense near the boundary, we have to replace it by something different. The geodesic collar is the natural substitute.

To give the local definition of bounded geometry, we have to explain which charts we want to use:

Definition 2. Choose 0 < r_{i}^{C} < r_{inj}(∂M). Choose x′ ∈ ∂M and an ONB of T_{x′}∂M. Define normal collar coordinates

κ_{x′} : \underbrace{B(0, r_{i}^{C})}_{⊂ℝ^{m−1}}×[0, r_{C}) → M : (v, t) ↦ exp_{exp_{x′}^{∂M} (v)}^{M}(tν).

(We compose exponential maps of ∂M and of M.) We have identified ℝ^{m−1} with T_{x′}∂M via an orthonormal frame and ν is the inward unit normal vector field. For x ∈ M − N_{1/3} the exponential map yields Gaussian coordinates

κ_{x} : B(0, r_{i}) → M : v ↦ exp_{x}^{M}(v)

(identify T_{x}M with ℝ^{m} via an orthonormal base). We denote normal collar coordinates and Gaussian coordinates with the common name normal coordinates and the range κ_{x}(B(0, ϵ)) (if x ∈ M−N_{1/3}) as well as κ_{x}(B(0, ϵ)× [0, r_{C})) (if x ∈ ∂M) with NR(x, ϵ).

Our local definitions of bounded geometry is the following:

Definition 3. A Riemannian manifold M has (local) bounded geometry if and only if (N), (IC), (I) of Definition 1 hold and (instead of (B))

(B1) For any K ∈ ℕ a constant C_{K} > 0 exists such that in arbitrary normal coordinates (if r_{C}, r_{i} and r_{i}^{C} are chosen sufficiently small) the following holds for the metric tensor g_{ij} and its inverse g^{ij}:

|D^{α}g_{ij}| ≤ C_{K} and |D^{α}g^{ij}| ≤ C_{K} ∀ |α| ≤ K.

It suffices to check the condition for an atlas.

The main result of the paper is the following:

Theorem 1. Let M be a Riemannian ∂-manifold. M has (local) bounded geometry if and only if it has (global) bounded geometry. In particular, we can drop the prefix in notation.

One implication follows directly form the formulas for R, l and their covariant derivatives in terms of g_{ij}, g^{ij} and their partial derivatives. The rest of this section is used to prove the other implication. We use the notation introduced above.

More precisely, we want to prove:

Theorem 2. Let M be a Riemannian manifold with boundary ∂M. Fix k ∈ ℕ. Suppose that |∇^{j}R| ≤ C and | ¯∇^{j}l| ≤ C for j ≤ k. Suppose r_{i}, r_{i}^{C}, r_{C} are nonnegative and sufficiently small (depending on C). Then there exists M-universal D > 0 so that in normal coordinates |D^{α}g_{ij}(x)| ≤ D and |D^{α}g^{ij}(x)| ≤ D for every partial derivative with |α| ≤ k. D depends only on C and r_{i}, r_{i}^{C}, r_{C}.

We prove this theorem in four steps: If k = 0, the Rauch comparison theorem yields the statement. Secondly, we prove the statement for Gaussian coordinates. Here we use Step 1. In the third step, we show that the boundary has the same properties as M: | ¯∇^{j}R̄| is bounded for j ≤ k. Last, we derive the statement of the theorem for normal boundary coordinates.

Step 1: Boundedness of g_{ij}

Proposition 1. Suppose we are in the situation of Theorem 2 and k = 0. Suppose x_{i} is a normal coordinate system. Then

C_{1} ≤ |∑ λ_{i} \frac{∂}{∂x_{i}} |_{T M} ≤ C_{2},  if ∑ λ_{i}^{2} = 1.  (1)

The constants C_{1,2} depend only on r_{i}, r_{C} and C. Particularly, the components g_{ij} and g^{ij} of the metric tensors in normal coordinates are bounded.

Proof. The last statement follows directly from the inequality (1). For this inequality, we apply Warner’s generalization of the Rauch comparison theorem [9, 4.3]. We compare with two complete manifold of constant sectional curvature −C and C, respectively. To compare with normal collar coordinates, choose a hypersurface in this manifold so that all the eigenvalues of its second fundamental form at one (comparison) point are equal to C in the first case and to −C in the second case. Inequality (1)is just the statement of the comparison theorem with C_{1,2} depending only on r_{i} or r_{C} and the manifold we compare with (i.e. on C). Here, r_{C} and r_{i} must be sufficiently small (depending only on the manifold we compare with, i.e. on C). The comparison theorem says nothing about the radial tangent vectors ∂_{r} = ∑ x_{i}∂_{i} (in Rauch’s version) and about normal tangent vectors ∂_{n}. But for these vectors Euclidian length and length in T M as well as the orthogonal complements coincide by Proposition 2. Therefore, the inequality is true in general. ⊓ ⊔

We used here (and at several other places) the Gauss lemma:

Proposition 2. Let (M, g) be a Riemannian manifold and exp : B(0, R) → M a Gaussian chart. Pull the metric g back to B(0, R). Then g(∂_{r}, ∂_{r}) = r^{2}, if ∂_{r} = ∑_{i} x_{i}∂_{i} is the radial vector field, and g(∂_{r}, v) = 0 if and only if v is tangent to the sphere with center 0 and through the base of v. Let K : ∂M × [0, r_{C}) → M be the Riemannian collar and pull g back to ∂M ×[0, r_{C}). Then g(∂_{t}, ∂_{t}) = 1 and g(∂_{t}, v) = 0 if and only if v is tangent to the corresponding translate ∂M × {t}.

Proof. Compare Gallot/Hulin/Lafontaine [4, 2.93] — the proof works also for the collar with no more than the obvious changes. ⊓ ⊔

Step 2: Gaussian coordinates. Now we will state a (differential) equation for the metric tensor g_{ij} in terms of the curvature tensor, so that a bound on partial derivatives of the components of the curvature tensor will give corresponding bounds for the metric. Partial and covariant derivative are related by the Christoffel symbols, so we will compute them, too.

Let p ∈ M be an interior point, so that Gaussian coordinates x with radius r_{i} around p exist. Choose an orthonormal base (ONB) {s_{i}} for T_{p}M. Using parallel transport along geodesics emanating from p, we construct a synchronous orthonormal frame {s_{i}(x)} of the tangent space restricted to the chart neighborhood. Let {θ^{i}} be the frame of 1-forms dual to {s_{i}} (therefore orthonormal). Now consider the connection forms θ_{j}^{i} for this frame defined by

∇s_{j} = ∑_{i} θ_{j}^{i}s_{i}

with associated Christoffel symbols Γ_{jk}^{i} and curvature tensor R_{jkl}^{i} given by

\begin{aligned}θ_{j}^{i} & = ∑_{k} Γ_{jk}^{i}dx_{k} and \\ dθ_{j}^{i} − ∑_{k} θ_{k}^{i} ∧ θ_{j}^{k} & = ∑_{kl} R_{jkl}^{i}dx_{k} ∧ dx_{l}. \end{aligned}

Additionally, we can express the curvature entirely in terms of s_{i} and θ^{i}:

dθ_{j}^{i} − ∑_{k} θ_{k}^{i} ∧ θ_{j}^{k} = ∑_{kl} K_{jkl}^{i}θ^{k} ∧ θ^{l}.

Define functions a_{j}^{i} and b_{j}^{i} via the equations

θ^{i} = ∑_{j} a_{j}^{i}dx_{j}; dx_{i} = ∑_{j} b_{j}^{i}θ^{j}.

\begin{aligned}Then R_{jkl}^{i} & = ∑_{αβ} K_{jαβ}^{i}a_{k}^{α}a_{l}^{β}  and \\ g_{ij} & = ∑_{α} a_{i}^{α}a_{j}^{α};  g^{ij} = ∑_{α} b_{α}^{i}b_{α}^{j}. \end{aligned}  (2)

Especially, the universal bounds on g_{ij} and g^{ij} from Proposition 1 yield bounds on a_{j}^{i} and b_{j}^{i}. Later we will also need the covariant differentials of ∂/∂x_{i} =: ∂_{i}:

∇_{∂_{k}}∂_{j} = ∑_{i} Γ̃_{jk}^{i}s_{i}.

Since ∂_{j} = ∑_{α} a_{j}^{α}s_{α},

Γ̃_{jk}^{i} = ∂_{k}a_{j}^{i} + ∑_{α} a_{j}^{α}Γ_{α_{k}}^{i}.  (3)

Let R be the radial vector field R(x) = ∑_{i} x_{i}∂/∂x_{i}. Then Atiyah-Bott-Patodi [1, a7] derive the following equations:

RΓ_{jk}^{i} + Γ_{jk}^{i} = ∑_{l} 2x_{l}R_{jkl}^{i}  ∀i, j, k;  (4)

(R^{2} + R)a_{l}^{i} = −2 ∑_{jk} R_{jkl}^{i}x_{j}x_{k}  ∀i, l.  (5)

Set f_{x}(t) = tΓ_{jk}^{i}(tx). Then

f′_{x}(t) = Γ_{jk}^{i}(tx) + t ∑_{l} x_{l}∂_{l}Γ_{jk}^{i}(tx) =^{(4)} ∑_{l} 2tx_{l}R_{jkl}^{i}(tx).

Therefore

Γ_{j}^{i}k(x) = f_{x}(1) = ∫_{0}^{1} ∑_{l} 2τx_{l}R_{jkl}^{i}(τx) dτ;

D_{x}^{α}Γ_{jk}^{i}(x) = ∫_{0}^{1} τ^{|α|} (D_{x}^{α}(x ↦ ∑_{l} x_{l}R_{jkl}^{i}(x))) (τx) dτ. (6)

Set f_{il}(t, x) = a_{l}^{i}(t·x). If′ denotes t-differentiation, tf_{ij}′(t, x) = Ra_{j}^{i}(tx), and t^{2}f′′(t, x) + tf′(t, x) = R^{2}a(tx). Equation (5) implies now

t^{2}f′′_{il} + 2tf′_{il} = −2t^{2} ∑_{kl} R_{jkl}^{i}(tx)x_{j}x_{k}.

With w(t, x) = t^{2}f′(t, x) we get w′ = t^{2}f′′ + 2tf′. Since w(0) = 0,

t^{2}f′_{il}(t, x) = −2 ∫_{0}^{t} τ^{2} ∑_{jk} R_{jkl}^{i}(τx)x_{j}x_{k} dτ ⟹^{τ=tu}

f′_{il}(t, x) = −2t ∫_{0}^{1} u^{2} ∑_{jkαβ} K_{jαβ}^{i}(tux)a_{k}^{α}(tux)a_{l}^{β}(tux)x_{j}x_{k} du. (7)

Now we are in the position to explain how the bounds on R and its covariant derivatives up to order k imply bounds on g_{ij}, g^{ij} and their partial derivatives up to order k. First, observe that instead of the metric tensor we can consider a_{j}^{i} and b_{j}^{i}. Moreover, the case k = 0 is done by Proposition 1.

Lemma 1. Let A, B be matrix functions which are inverse to each other. Therefore,

\frac{∂}{∂x_{i}} B = \frac{∂}{∂x_{i}} A^{−1} = −A^{−1}(\frac{∂}{∂x_{i}} A)A^{−1} = −B(∂_{i}A)B

Iterated application of this rule and the product rule yields

D_{x}^{α}B = P_{α}(B, D_{x}^{β}A; β ≤ α)

where P_{α} is a well defined non-commutative polynomial. Therefore, bounds for the partial derivatives of A up to order k and on B immediately yield bounds for the partial derivatives of B.

Lemma 1 applies to the matrices A = (a_{j}^{i}) and B = (b_{j}^{i}). Moreover, by Proposition 1, we have a universal bound for B. It only remains to find bounds for the derivatives of (a_{j}^{i}). To do this we have to relate the covariant derivatives of R with partial derivatives of K_{jkl}^{i}.

Lemma 2. There is a polynomial P_{α,ijkl} (only depending on α, i, j, k, l) in partial derivatives up to order (|α| − 1) of K_{∗∗∗}^{∗}, Γ_{∗∗}^{∗}, and a_{∗}^{∗} s.t.

∇_{∂_{i_{1}}} … ∇_{∂_{i_{k}}} R(s_{i}, s_{j}, ∂_{k}, ∂_{l}) =^{α=(i_{1},...,i_{k})} D^{α}K_{jkl}^{i} + P_{α,ijkl}.

Proof. This follows from the well known formula for covariant differentials in coordinates. Note that for |α| = 1 only Γ_{jk}^{i} shows up, since K_{jkl}^{i} are the components of R entirely in terms of s_{i}. But if we iterate the covariant differentials, we have to take into account that we contracted ∇R with ∂_{i} and not with s_{i}. This yields (via ∇∂_{i}) Γ̃_{jk}^{i} and, since we iterate the covariant differentials, their partial derivatives up to order |α|−2. Since Γ̃_{jk}^{i} = ∂_{k}a_{i}^{j}+ ∑_{α} a_{i}^{α}Γ_{jk}^{α}, the result follows. ⊓ ⊔

Now we give a bootstrap argument as follows: By assumption the curvature is bounded. Therefore, since {s_{i}} is orthonormal by construction, K_{jkl}^{i} are uniformly bounded. We know already that the same is true for a_{j}^{i}.

Assume by induction that for some r ≥ 0 we have found universal bounds on the partial derivatives up to order r of K_{jkl}^{i} and a_{j}^{i} and on the derivatives up to order r − 1 of Γ_{jk}^{i}. Suppose we have bounds on |R| , …, |∇^{r+1}R|.

From equation (2), relating K_{jkl}^{i} and R_{jkl}^{i}, we get universal bounds of the partial derivatives up to order r of R_{jkl}^{i}. Then Equation (6)yields universal bounds for the derivatives of order r of Γ_{jk}^{i}. Lemma 2 and the bound on ∇^{r+1}R yield universal bounds on (r + 1)-order partial derivatives of K_{jkl}^{i} (use that the lengths of ∂_{i} are universally bounded by Proposition 1).

Now remember the following equation (7):

f′_{il}(t, x) = −2t ∫_{0}^{1} u^{2} ∑_{jkαβ} (K_{jαβ}^{i}a_{k}^{α}a_{l}^{β}) (tux)x_{j}x_{k} du.

Let α be a multi-index with |α| = r + 1. We differentiate the equation with respect to x to get an equation for D_{x}^{α}f(t, x) = t^{|α|}(D^{α}a)(tx). This yields

(D_{x}^{α}f_{il})′(t, x) = −2t ∫_{0}^{1}(u^{2} ∑_{jkβγ} K_{jβγ}^{i}(tux)((D_{x}^{α}f_{βk})f_{γ,l} +

+ (D^{α}f_{γ,l})f_{βk})(tu, x)x_{j}x_{k} du) − 2t ∫_{0}^{1} P_{α} du. (8)

Here P_{α} is a polynomial in t, u, x, partial derivatives up to order (r + 1) of K_{∗∗∗}^{∗} at tux, and partial derivatives up to order r of f_{∗∗} at (tu, x). The induction hypothesis yields for 0 ≤ t ≤ 1 the inequality

|(D_{x}^{α}f)′(t, x)| ≤ C_{1} sup_{0≤τ≤t}{|D_{x}^{α}f(τ, x)|} + C_{2},  (9)

where C_{1,2} > 0 can be expressed in the universal bounds of the induction hypothesis. Moreover, note D^{α}f(0, x) = 0 (since |α| ≥ 1).

Lemma 3. Let h(t) be the unique solution of h′(t) = C_{1}h(t) + C_{2} with h(0) = 0. A Taylor series ansatz easily shows that h is a positive monotonous increasing function, with an explicit bound h(t) ≤ C = C(C_{1}, C_{2}) for 0 ≤ t ≤ 1. Abbreviate g(t) := D_{x}^{α}f(t, x). The inequality (9) implies |g(t)| ≤ h(t). Especially

|D^{α}a_{j}^{i}(x)| = |g_{j}^{i}(1)| ≤ h(1) ≤ C

Proof. We only must show |g(t)| ≤ h(t). Let h_{n} be the unique solution of

h′_{n}(t) = C_{1}h_{n}(t) + C_{2} + 1/n.

Then h_{n}(t) →^{n→∞} h(t) uniformly for 0 ≤ t ≤ 1. Therefore, it suffices to show |g(t)| ≤ h_{n}(t). For a contradiction, assume that for some n and t |g(t)| > h_{n}(t). Set t_{0} := inf_{0≤t}{|g(t)| > h_{n}(t)}. Then |g(t_{0})| = h(t_{0}) (since g(0) = 0 = h(0)). h_{n} is monotonous and |g(t)| ≤ h_{n}(t) ∀t ≤ t_{0}. Consequently, sup_{t≤t_{0}} |g(t)| = |g(t_{0})|. Then inequality (9) shows

|g′(t_{0})| ≤ C_{1} |g(t_{0})| + C_{2} < h′_{n}(t_{0}).

Moreover, d/dt |g(t_{0})| ≤ |g′(t_{0})| (if g(t_{0}) = 0 (d/dt is understood to be the right derivative): compare [5, III.3.2]). It follows |g(t)| < h_{n}(t) for t ∈ [t_{0}, t_{0} + ϵ) and ϵ > 0 sufficiently small. But this contradicts the choice of t_{0}. ⊓ ⊔ Step 3: Curvature of ∂M. Let R̄ be the curvature (0, 4)-tensor on ∂M, R the curvature tensor of M, and l the second fundamental form, considered as (0, 2)-tensor using the inward unit normal vector field ν.

In the following we apply the covariant differential ¯∇ of ∂M to (0, q)- tensors on M. We implicitly assume that we first restrict the tensor to ∂M.

We compute the covariant derivatives ¯∇^{k}R̄ using the following rules:

Lemma 4. Suppose T is a (0, q)-tensor on M, S a (0, p)-tensor on ∂M, and S^{∗_{1}} the (1, p − 1)-tensor on ∂M given by g(S^{∗_{1}}(v_{2}, …, v_{p}), v_{1}) = S(v_{1}, …, v_{p}). Let σ be a permutation (operating on a multiple tensor product by permutation of the factors) with σ^{−1}(1) ≤ p. c denotes the (unique) contraction of a (0, r)-tensor with a (1, s)-tensor. We adopt the general convention ∇ : C^{∞}(E) → C^{∞}(E ⊗ T^{∗}M). Then the following holds:

1. ¯∇T = ∇T −∑_{i} c(T ⊗l∘σ_{i}, ν), where σ_{i} are appropriate permutations.

2. ¯∇((T ⊗ S) ∘ σ) = (( ¯∇T) ⊗ S) ∘ σ + (T ⊗ ¯∇S) ∘ σ.

3. ¯∇c((T ⊗ S) ∘ σ, ν) = c((∇T) ⊗ S ∘ σ′, ν) + c(T ⊗ ¯∇S ∘ σ, ν) + ∑_{i} c(c(T ⊗S ∘σ, ν)⊗l∘σ_{i}, ν)+c(T ⊗S ∘σ, l^{∗_{1}}), σ′ and σ_{i} appropriate permutations.

4. ¯∇c(T, ( ¯∇^{k}l)^{∗_{1}}) = c( ¯∇T, ( ¯∇^{k}l)^{∗_{1}}) + c(T, ( ¯∇^{k+1}l)^{∗_{1}}).

Corollary 1. Iterated application of the rules of Lemma 4 shows that ¯∇^{k}R̄ is a finite sum of tensor products and possibly iterated contractions, composed with permutations, involving (i) ∇^{j}R for j ≤ k; (ii) ¯∇^{j}l for j < k; (iii) ( ¯∇^{j}l)^{∗_{1}} for j < k − 1; and (iv) ν. In particular if there are bounds for the building blocks (i) and (ii), this yields a concrete bound for ¯∇^{k}R̄ (since tensor products and contractions of bounded tensors remain bounded and permutations are isometric. Note |ν| = 1 and |S^{∗_{1}}| = |S|).

Proof (of Lemma 4). Formulas (ii) and (iv) are well known. For (i):

¯∇T (v_{1}, …, v_{p}, X)

= X.T (v_{1}, …, v_{p}) − T ( ¯∇_{X}v_{1}, …, v_{p}) − ⋯ − T (v_{1}, …, ¯∇_{X}v_{p})

=^{¯∇_{X}Y =∇_{X}Y −l(X,Y)ν} X.T (v_{1}, …, v_{p}) − T (∇_{X}v_{1}, …) − …

+ T (ν, v_{2}, …)l(v_{1}, X) + ⋯ + T (v_{1}, …, v_{p−1}, ν)l(v_{p}, X)

= ∇T (v_{1}, …, v_{p}, X) + ∑_{i} c(T ⊗ l ∘ σ_{i}, ν)(v_{1}, …, v_{p}, X).

For (iii):

¯∇_{X}c(T ⊗ S ∘ σ, ν)(v_{2}, …, v_{p+q}) =^{v_{1}:=ν}

= (X.T (v_{σ1}, …, v_{σp}))S(v_{σ(p+1)}, …, v_{σ(p+q)})

+ T (v_{σ1}, …, v_{σp})(X.S(v_{σ(p+1)}, …, v_{σ(p+q)}))

− ∑_{\begin{matrix}i=1 \\ σ(i)≠1 \end{matrix}}^{p+q} T ⊗ S(v_{σ1}, …, ¯∇_{X}v_{σi}, …, v_{σ(p+q)})

= c(T ⊗ ¯∇_{X}S ∘ σ, ν)(v_{2}, …, v_{p+q})

+ (X.T (v_{σ1}, …, v_{σp}) − ∑_{i=1}^{p} T (v_{σ1}, …, ∇_{X}v_{σi}, …, v_{σp})) S(…)

− ∑_{\begin{matrix}i=1 \\ σ(i)≠1 \end{matrix}}^{p} \underbrace{T (v_{σ1}, …, l(X, v_{σi})ν, …, v_{σp})}_{=T (···)l(X,v_{σi})} S(…)

+ T (v_{σ1}, …, ∇_{X}ν, …, v_{σp})S(…) = c(T ⊗ ¯∇_{X}S ∘ σ, ν)(X, …) + c((∇_{X}T) ⊗ S ∘ σ, ν)(…)

− ∑_{i} c(c(T ⊗ S ∘ σ, ν) ⊗ l ∘ σ_{i}, ν)(…) + c(T ⊗ S ∘ σ, ∇_{X}ν)(…).

Now, if Y ∈ C^{∞}(T ∂M) then

0 = X.g(ν, Y) = g(∇_{X}ν, Y) + g(ν, ∇_{X}Y)
⟹ g(∇_{X}ν, Y) = l(X, Y) = l(Y, X)
0 = X.g(ν, ν) = 2g(∇_{X}ν, ν).
Together: ∇_{X}ν = l^{∗_{1}}(X) ⟺ ∇ν = l^{∗_{1}}.

This finishes the proof. ⊓ ⊔

Step 4: Normal boundary coordinates. First consider ∂M as a Riemannian (m − 1)-dimensional manifold of its own. Let ν be the unit inward normal vector field. Step 3 shows that bounds on the covariant derivatives ∇^{j}R of the curvature of M, and ¯∇^{j}l of the second fundamental form of ∂M (0 ≤ j ≤ k) give rise to bounds on the derivatives ¯∇^{j}R̄ (0 ≤ j ≤ k) of the curvature of ∂M. Therefore we can use step two to construct the orthonormal frame {s_{i}}_{1≤i≤m−1} of T ∂M. Extend this to an orthonormal frame of T M|_{∂M} by setting s_{m} = ν. By parallel transport along geodesics with initial speed ν we get a synchronous orthonormal frame of T M on a normal collar neighborhood. Define the dual frame {θ^{i}}, the Christoffel symbols Γ_{jk}^{i} and Γ̃_{jk}^{i}, the curvature coefficients R_{jkl}^{i} and K_{jkl}^{i}, and the base change matrices a_{j}^{i} and b_{j}^{i} in exactly the same way as we did in Step 2. Note that the fundamental relations (2) and (3) relating R_{jkl}^{i} and K_{jkl}^{i}, Γ̃ and Γ as well as g_{ij} and a_{j}^{i} remain true.

Now we come to the differential equations which relate these quantities. Let x_{1}, …, x_{m} be the normal coordinates and x_{m} the normal direction, i.e. ∂_{m}|_{∂M} = ν. Our approach is similar to the one in [1]. By construction {s_{i}} is parallel to ∂_{m}. This translates to

c(∂_{m})θ_{j}^{i} = 0, i.e. Γ_{jm}^{i} = 0  ∀i, j  (10)

(c(∂_{m}) denotes contraction with ∂_{m}). The Lie derivative along ∂_{m} (denoted by ∂_{m}) acts on differential forms via ∂_{m} = c(∂_{m})d + dc(∂_{m}). Therefore,

\begin{aligned}∂_{m}θ_{j}^{i} & = c(∂_{m})dθ_{j}^{i} = c(∂_{m})(dθ_{j}^{i} − ∑_{k=1}^{m} θ_{k}^{i} ∧ θ_{j}^{k}) \\ & = (∂_{m})(∑_{k,l} R_{jkl}^{i})dx_{k} ∧ dx_{l} = 2 ∑_{k} R_{jmk}^{i}dx_{k}. \end{aligned}

On the other hand ∂_{m}θ_{j}^{i} = ∑_{k} ∂_{m}(Γ_{jk}^{i})dx_{k}.

Equating coefficients and applying D_{x}^{α} yields

∂_{m}(D^{α}Γ_{jk}^{i}) = 2D^{α}R_{jmk}^{i} ∀i, j, k.  (11)

Additionally we need an equation for a_{j}^{i}. We apply ∂_{m} twice to the dual frame. Since ∂_{m} = s_{m}, c(∂_{m})θ^{i} = δ_{im}s (δ is the Kronecker symbol). Then

∂_{m}θ^{i} = c(∂_{m})dθ^{i} + dc(∂_{m})θ^{i} = c(∂_{m})dθ^{i}.

The connection is torsion free. This means

\begin{matrix}dθ^{i} & = ∑_{j} θ_{j}^{i} ∧ θ^{j} \\ ⟹ & ∂_{m}(θ^{i}) = ∑_{j}(∂_{m})θ_{j}^{i} ∧ θ^{j} = −θ_{m}^{i} \\ ⟹ & ∂_{m}^{2}(θ^{i}) = −∂_{m}(θ_{m}^{i}) = −2 ∑_{k} R_{mmk}^{i}dx_{k}. \end{matrix}

The left hand side can be computed in terms of a_{j}^{i}

∂_{m}(θ^{i}) = ∑_{j} ∂_{m}(a_{j}^{i})dx_{j} ⟹ ∂_{m}^{2}(θ^{i}) − ∑_{j} ∂_{m}^{2}(a_{j}^{i})dx_{j}.

Equating coefficients and computing R_{jkl}^{i} in terms of K_{jkl}^{i} yields

∂_{m}^{2}a_{j}^{i} = −2 ∑_{αβ} K_{mαβ}^{i}a_{m}^{α}a_{j}^{β}  ∀i, j.  (12)

To make use of the differential equation (11) and (12) we have to supply the initial conditions (x_{m} = 0). For a_{j}^{i} this is very easy: if i, j < m, a_{j}^{i}|_{x_{m}=0} is given by application of Step 2 to ∂M. In particular, we get universal bounds for these functions. And by construction a_{m}^{i} = a_{i}^{m} = δ_{im}. For the first derivative we have ∂_{m}a_{j}^{i}|_{x_{m}=0} = −Γ_{jm}^{i} (this follows from the two formulae for ∂_{m}θ^{i}).

It remains to compute Γ_{jk}^{i} on ∂M. By definition ¯∇_{∂_{i}}s_{j} = ∇_{∂_{i}}s_{j} − l(∂_{i}, s_{j})ν for i, j < m. If we define l_{j}^{i} := l(∂_{i}, s_{j}) then for j, k < m

Γ_{jk}^{i}|_{x_{m}=0} = \begin{cases}Γ̄_{jk}^{i}|_{x_{m}=0}; & i < m \\ l_{j}^{i}; & i = m. \end{cases}

We have already seen that Γ_{jm}^{i} = 0 ∀i, j, since {s_{i}} is synchronous along ∂_{m}. To compute Γ_{mj}^{i} we need

∇_{∂_{j}}∂_{m} = ∇_{∂_{m}}∂_{j} = ∑_{i} l_{j}^{i}s_{i}.

The first equation is due to the fact that [∂_{m}, ∂_{i}] = 0. The second one is stated in Warner [9] and is a consequence of ∂_{m} ⊥ ∂_{i} for i < m all over the collar (the latter follows from the Proposition 2). It follows for i < m

Γ_{mj}^{i} = \begin{cases}l_{j}^{i}; & j < m \\ 0; & j = m. \end{cases}

The arguments of Step 2 show that the initial functions and their partial derivatives up to order k in directions x_{1}, …, x_{m−1} are uniformly bounded, if bounds exist on the covariant derivatives up to order k of the second fundamental form and of R (hence by Step 3 also on R̄).

We proceed with a bootstrap argument similar to the argumentation in Step 2: First, K_{jkl}^{i} and (since the length of ∂_{i} is bounded by Proposition 1) R_{jkl}^{i} are M-uniformly bounded functions. From Equation (11) we find bounds for Γ_{jk}^{i}. With these, the bound on ∇R translates to bounds on ∂_{α}K_{jkl}^{i}. From (12) it follows that ∂_{α}a_{j}^{i} are M-universally bounded. This, together with the relation (2) between R_{jkl}^{i} and K_{jkl}^{i} yields bounds on ∂_{α}R_{jkl}^{i}. Now we start again with (11) to get bounds on the first order derivatives of Γ_{jk}^{i}. Inductively, this concludes the proof of Theorem 2.

3. Technical properties of manifolds of bounded geometry

We look at the volume of balls and the change of (geodesic) variable functions on manifolds of bounded geometry now, and we produce the atlas and partition of unity as described above

Lemma 5. Let M be a ∂-manifold of bounded geometry. We find r > 0 so that for every x ∈ M the metric ball B(x, r) is contained in some normal chart NR(y, ϵ). Moreover, 0 < D_{1}(r) < D_{2}(r) exist for any such r so that

D_{1}(r) ≤ vol(B(x, r)) ≤ D_{2}(r) ∀x ∈ M,
D_{1}(r) ≤ vol(NR(x′, r)) ≤ D_{2}(r) ∀x′ ∈ ∂M, and
D_{1}(r) ≤ vol(B(x′, r) ⊂ ∂M) ≤ D_{2}(r) ∀x′ ∈ ∂M.

Proof. Boundedness implies the existence of C_{1}, C_{2} > 0 so that in normal coordinates

∥(g^{ij})_{ij}∥ < C_{1}, ∥(g_{ij})_{ij}∥ < C_{1}, and C_{2} ≤ \sqrt{|det g_{ij}|} ≤ C_{1}.

The first assertion is trivial for x ∈ M − N_{1/3}, if we choose r ≤ r_{i} (r_{i} of the definition of bounded geometry). For x ∈ N_{1/3}, project x along the geodesic collar to x′ ∈ ∂M and consider NR(x′, r_{inj}). Because of the boundedness of the norm of the matrices (g^{ij})_{ij} and (g_{ij})_{ij}, every path which starts from x and does leave NR(x′, r_{inj}) has length at least r for some M-universal constant r > 0. This proves the first statement. The assertion about the volumes follows from the upper and lower bounds of \sqrt{|det g_{ij}|}. (We have to integrate these functions over a fixed cylinder and ball, respectively, in Euclidian space.) It remains to explain why we can find volume bounds for the metric balls if they do not lie in the range of a Gaussian chart. In this case we can suppose that B(x, r) ⊂ NR(x′, ϵ) for some cylinder NR(x′, ϵ). We can not describe the image B = B(x, r) in terms of these coordinates. However, from the bounds on the metric tensor we know that Muniversal r_{1}(r) and r_{2}(r) exist so that (if we equip the cylinder NR(x′, ϵ) with the Euclidian metric)

B_{Eucl.}(x, r_{1}) ⊂ B ⊂ B_{Eucl.}(x, r_{2})

The volume of B_{Eucl.}(x, r_{1}) in (M, g) is M-universally bounded by the same reason as above. This concludes the proof. ⊓ ⊔

The following is a very important technical lemma to do analysis on manifolds of bounded geometry.

Proposition 3. (Partition of unity) Let M be a manifold of bounded geometry. There is r_{m} > 0 so that if 0 < r < r_{m} then a countable covering of M exists by sets {NR(x_{i}, r)}_{i∈ℤ} which has the following properties:

– x_{i} ∈ ∂M for i ≥ 0 and x_{i} ∈ M − N_{2/3} for i < 0.

– There is M_{f} so that ∀s < r_{m} and ∀x NR(x, s) ∩ NR(x_{i}, r) ≠ ∅ for at most M_{f} of the x_{i}.

– {NR(x_{i}, r/2)}_{i∈ℤ} is a covering of M, too.

Denote with κ_{i} : B(0, r) → NR(x_{i}, r) (i < 0) and κ_{i} : B(0, r) × [0, r_{C}) → NR(x_{i}, r) for i ≥ 0 the corresponding charts (r_{C} must be sufficiently small).

To this covering, a subordinate partition of unity {φ_{i}} exists which is uniformly bounded, i.e. ∀K ∈ ℕ we find C_{K} > 0 so that

|D^{α}φ_{i}| ≤ C_{K}  ∀i ∈ ℤ ∀ |α| ≤ K  (in normal coordinates).

We get the following data:

– a covering NR(x_{i}, r)

– with charts κ_{i} : ℝ^{m} → NR(x_{i}, r)

– and a nice subordinate partition of unity φ_{i} (i ∈ ℤ).

Proof. If ∂M = ∅, compare Shubin [8, A1.2 and A1.3] for the first statement. We extend his argument to cover our more general situation. Set r_{m} = min{r_{i}/3, r_{i}^{C}/3}. Let 0 < r < r_{m}. Now choose a maximal set of points {x_{i} ∈ ∂M}_{i∈ℕ} so that (in ∂M) all {B(x_{i}, r/4)} are disjoint. Similarly, choose a maximal set of points {x_{i} ∈ M−N_{2/3}}_{i<0} so that all B(x_{i}, r/4)_{i<0} are disjoint. Then

⋃_{i<0} NR(x_{i}, r/2) = ⋃_{i<0} B(x_{i}, r/2)} covers M − N_{2/3}.

Else we find z ∈ M − N_{2/3} which has distance ≥ r/2 to all of the x_{i}. Then B(z, r/4) ∩ B(x_{i}, r/4) = ∅ ∀i < 0, violating the maximality of the set {x_{i}}_{i<0}. Similarly, {B(x_{i}, r/2)}_{i≥0} covers ∂M ⟹ {NR(x_{i}, r/2)}_{i≥0} covers the collar N. Now we have to show that the covering {NR(x_{i}, r/2)}_{i∈ℤ} has finite covering dimension. For this, choose 0 < ϵ < 3r_{m}. Lemma 5 shows that we find 0 < D_{1}(ϵ) < D_{2}(ϵ) so that

D_{1}(ϵ) ≤ vol(NR(x, ϵ)) ≤ D_{2}(ϵ) ∀x ∈ M and
D_{1}(ϵ) ≤ vol(B(x′, ϵ) ⊂ ∂M) ≤ D_{2}(ϵ) ∀x′ ∈ ∂M.

Choose z ∈ M. If z ∈ N_{1/3} then z ∉ NR(x_{i}, r) ∀i < 0 if we choose r_{i} sufficiently small (as we can do). Project z along the geodesic collar to z′ ∈ ∂M. If z′ ∈ B(x_{i}, r) ⊂ ∂M then B(x, r/4) ⊂ B(z′, 2r) ⊂ ∂M (2r < 3r_{m} ≤ r_{i}). For the number N_{1} of these x_{i} we have, because all B(x_{i}, r/4) ⊂ ∂M (i ∈ ℕ) are disjoint:

N_{1} ≤ \frac{sup_{z′∈∂M} vol(B(z′, 2r) ⊂ ∂M)}{inf_{x∈∂M} vol(B(x, r/4) ⊂ ∂M)} ≤ \frac{D_{2}(2r)}{D_{1}(r/4)}.

Note z ∈ NR(x_{i}, r) ⟺ z′ ∈ B(x_{i}, r) ⊂ ∂M (i ≥ 0). If z ∈ M −N_{1/3}, it is (by the same argument) contained in at most N_{1} cylinders of type NR(x_{i}, r)_{i≥0}. Now, it is also possible that z ∈ NR(x_{i}, r) = B(x_{i}, r) (i < 0). Then B(x_{i}, r/4) ⊂ B(z, 2r). For the number N_{2} of these x_{i} we have again N_{2} ≤ D_{2}(2r)D_{1}(r/4)^{−1}. Therefore, {NR(x_{i}, r)}_{i∈ℤ} is a covering of finite covering dimension.

Similarly to the proof of Lemma 5 we show that r_{m} > 0 exists so that for

x ∈ im(\underbrace{B(y′, r_{i}/2)}_{⊂∂M} ×[0, r_{C}/2)) holds: B(x, r_{m}) ⊂ im(\underbrace{B(y′, r_{i})}_{⊂∂M} ×[0, r_{C}))

(image under the geodesic collar map). Replacing r_{C} by r_{C}/2 we can prove the existence of M_{f} just as we proved finite covering dimension.

It remains to construct the subordinate partition of unity. Choose a smooth cutoff function φ : ℝ^{m} → [0, 1] with φ(x) = 1 if |x| ≤ r/2 and φ(x) = 0 if |x| ≥ r. Denote the restriction to ℝ^{m−1} also with φ. Choose smooth ψ : ℝ → [0, 1] with ψ(x) = 0 if x ≥ r_{C} and ψ(x) = 1 if x ≤ 2r_{C}/3. Via the normal coordinates this yields cutoff functions f_{i} on NR(x_{i}, r) with f_{i}κ_{i}(y′, t) = φ(y′)ψ(t) if i ≥ 0 and f_{i}κ_{i} = φ if i < 0. Therefore, if κ is any normal chart, f_{i} ∘ κ = φ ∘ (κ_{i}^{−1}κ) (i < 0) and f_{i} ∘ κ = (φ · ψ) ∘ (κ_{i}^{−1}κ) (i ≥ 0). The chain rule shows that the bounds on derivatives up to order K of the coordinate changes (Proposition 4) yield bounds on the partial derivatives up to order K of f_{i} in normal coordinates. To construct the partition of unity, set F = ∑_{i∈ℤ} f_{i} (at each point there are at most N_{1} + N_{2} non-trivial summands). Since

M − N_{2/3} ⊂ ⋃_{i<0} NR(x_{i}, r/2) and N_{2/3} ⊂ ⋃_{i≥0} NR(x_{i}, r/2),

for each z ∈ M at least one of f_{i}(z) = 1 ⟹ F ≥ 1. Define

φ_{i} := f_{i}/F.

Obviously, {φ_{i}}_{i∈ℤ} is a smooth partition of unity subordinate to our covering. Pick one φ_{i} and one normal chart κ. For partial derivatives up to order K in normal coordinates observe

|D^{α}(φ_{i} ∘ κ)| = |D^{α} \frac{f_{i} ∘ κ}{F ∘ κ}|

= \frac{|P_{α}(D^{β}(f ∘ κ), D^{γ}(F ∘ κ); |β| , |γ| ≤ |α|)|}{|F ∘ κ|^{2^{|α|}}}

≤^{|F |≥1} |P_{α}(D^{β}(f ∘ κ), D^{γ}(F ∘ κ))| .

P_{α} is a polynomial entirely determined by α. At every point v, D^{γ}(F ∘κ)|_{v} is the sum of at most N_{1}+N_{2} summands of the type D^{γ}(f_{s}∘κ)|_{v}. Therefore, for all the entries of P_{α}, we have M-universal bounds. This yields a bound C for |D^{α}(φ_{i} ∘ κ)| if |α| ≤ K. ⊓ ⊔

Proposition 4. Suppose, M is a Riemannian ∂-manifold of bounded geometry. More precisely, suppose C > 0 is a bound for partial derivatives up to order k + 1 of the metric tensors g^{ij} and g_{ij} in normal coordinates. Then an M-universal D > 0 exists so that, if κ_{1} : U_{1} ⊂ ℝ^{m} → M and κ_{2} : U_{2} ⊂ ℝ^{m} → M are arbitrary normal charts, the following holds for f := κ_{1}^{−1} ∘ κ_{2} : U_{0} ⊂ ℝ^{m} → ℝ^{m} (U_{0} the natural domain of definition of the composition):

|D^{α}f| ≤ D ∀ |α| ≤ k.

The maps κ_{i} are solutions of certain ordinary differential equations (o.d.e.), namely, the equations for geodesics. Therefore, first we state a result about arbitrary differential equations.

Lemma 6. Let x′ = F (t, x) be a system of o.d.e. (t ∈ ℝ, x ∈ ℝ^{n}), F ∈ C^{∞}(ℝ × ℝ^{n}, ℝ^{n}). Suppose φ(t, x) is the flow of this equation. Then

|\frac{∂φ}{∂x_{k}} (t, x_{0})| ≤ exp(∥D_{x}F ∥ t) ∀t ≥ 0.  (13)

For higher derivatives, we find a universal expression Expr_{α}, only depending on α and involving iterated exponentials, so that

|D_{x}^{α}(x_{0}, t)| ≤ Expr_{α}(∥D_{x}^{β}F ∥ , t_{0} | β ≤ α).

Proof. The theory of o.d.e. [5, V.3.1.] tells us that we have the linear differential equation

α′(t) = \frac{∂F}{∂x} (t, φ(t, x)) · α(t); α(0) = e_{k} = (0, …, 1, …, 0)

for ∂_{k}φ(t, x). For linear differential equations, [5, IV.4.2] gives inequalities which directly imply (13).

Higher derivatives also fulfill a linear differential equation:

(D_{x}^{α}φ)′(x, t) = (D_{x}F)(t, φ(x, t)) · D_{x}^{α}φ(x, t) +

P_{α}(D_{x}^{γ}φ, (D_{x}^{β}F)(t, φ(x, t)); γ < α, β ≤ α) (14)

where P_{α} is a polynomial matrix which depends only on α. By induction and using the inequalities for solutions of such linear differential equations, the proposition follows. ⊓ ⊔

Corollary 2. Reduction of order implies that the corresponding statement holds for o.d.e. of order k.

Changes of normal coordinates

Corollary 3. Let p ∈ U ⊂ ℝ^{m}, g_{ij} a Riemannian metric on U and exp_{p} : B(r, 0) → U the exponential map at p (we identify T_{p}U with ℝ^{m} via an orthonormal frame). If r is sufficiently small then exp_{p} is a diffeomorphism onto some open subset of U, and the derivatives up to order k of exp_{p} are bounded in universal expressions containing g_{ij}, their derivatives up to order k + 1, and r.

Proof. We have exp_{p}(x) = φ(x, p, 1), where φ is the flow of the differential equation for geodesics (φ(x, q, 0) = q, φ′(x, q, 0) = x). Lemma 6 applies to this equation x′′ = F (x), and F (x) = − ∑_{ij} Γ_{ij}(x)x′_{i}x′_{j} is given by g_{ij} and its first order derivatives. ⊓ ⊔

Corollary 4. Suppose p ∈ U′ ⊂ ℝ^{m−1}, κ : U′ × [0, r_{C}) → V × [0, r_{C}) is a normal boundary chart on the Riemannian manifold V × [0, r_{C}) with metric g_{ij} (g_{in} = δ_{in} = g_{ni}). Then κ and its derivatives up to order k are bounded by a universal expression in g_{ij} and their derivatives up to order k + 1.

Proof. κ(q, s) = φ_{1}(s · ∂_{n}, φ_{2}(q, p, 1), 0, 1), where φ_{1} is the flow of the differential equation for the geodesics in V × [0, r_{C}) (φ_{1}(v, p′, s′, 0) = (p′, s′), φ′_{1}(v, (p′, s′), 0) = v), and φ_{2} is the flow of the differential equation for geodesics on V. κ is therefore the composition of two flows to which Lemma 6 applies exactly as in the previous corollary. ⊓ ⊔

Proof (of Proposition 4). By Theorem 2 we have universal bounds for g_{ij} and their derivatives up to order k + 1 in normal coordinates. To apply the preceding two corollaries we use one intermediate chart:

κ_{1}^{−1} ∘ κ_{2} = (κ_{1}^{−1} ∘ κ_{0}) ∘ (κ_{0}^{−1} ∘ κ_{2}).

This covers the following cases: κ_{2} Gaussian coordinates and κ_{1} arbitrary normal coordinates (then κ_{0} shall be a Gaussian chart around points in the common range of κ_{1} and κ_{2}), or κ_{1} and κ_{2} are both normal boundary coordinates (then their common range is U × [0, r_{C}) —in terms of the geodesic collar— with U ⊂ ∂M, and κ_{0} shall be a normal boundary chart centered at points of Uindependent. It is immediate that the preceding corollaries apply to κ_{0}^{−1} ∘ κ_{2} and to κ_{1}^{−1} ∘ κ_{0} and yield the desired bounds. The next proposition covers the remaining case κ_{1} a Gaussian chart and κ_{2} a normal collar chart and concludes the proof. ⊓ ⊔

Proposition 5. In the situation of Proposition 4, let f = κ_{1}^{−1} ∘κ_{2} : U → V with κ_{2} a normal boundary chart, κ_{1} a Gaussian chart. Let g = f^{−1}. We have found C > 0 of g so that |D^{α}g| ≤ C ∀ |α| ≤ k. Additionally, there is C′ independent of f so that |D^{α}f| ≤ C′ ∀ |α| ≤ k.

Proof. For the differentials holds Df = (Dg)^{−1}. By Lemma 1, a bound on Df together with the bounds for the derivatives of Dg (given by C) yields a bound C′ for the higher derivatives of f. Moreover, C′ only depends on k, C and the bound on Df. Now Df = Dx_{1} · Dx_{2}^{−1}. Take v ∈ T U with |v|_{ℝ^{m}} = 1 ⟹^{Proposition 1} |Dx_{2}^{−1}v|_{T M} ≤ C_{2}. Moreover (again by Proposition 1)

C_{1} ≤ \begin{matrix}|Dx_{2}^{−1}v| & M \\ |Dx_{1}Dx_{2}^{−1}v| & ℝ^{m} \end{matrix} ⟹ |Dx_{1}Dx_{2}^{−1}v|_{ℝ^{m}} ≤ \frac{|Dx_{2}^{−1}v|_{M}}{C_{1}} ≤ \frac{C_{2}}{C_{1}} ;

i.e. we have the bound C_{2}/C_{1} for ∥Df∥. The last computation concludes the proof of Proposition 4. ⊓ ⊔

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