We will use the following form of the inverse function theorem.
Theorem 2.7.1 (Inverse Function). Let U be an open subset of Rm and K a compact space. We suppose that ϕ : K × U → Rm is a continuous map such that
(1) for each k ∈ K, the map ϕk: U → Rm, x 7→ ϕ(k, x) is C1, (2) The map ∂ϕ∂x : K × U → L(Rm, Rm), (k, x) 7→ ∂ϕ∂x(k, x) is
continuous.
If C ⊂ U is a compact subset such that
(i) for each k ∈ K, and each x ∈ C, the derivative ∂ϕ∂x(k, x) is an isomorphism,
(ii) for each k ∈ K, the map ϕk is injective on C, then there exists an open subset V such that
(a) we have the inclusions C ⊂ V ⊂ U ,
(b) for each k ∈ K, the map ϕk induces a C1 diffeomorphism of V on an open subset of Rm.
Proof. Let k·k denote a norm on Rm, and let d denote its associ-ated distance. Let us show that there is an integer n such that, if we set
Vn=
x | d(x, C) < 1 n
,
then, the restriction ϕk|Vn is injective, for each k ∈ K. We argue by contradiction. Suppose that for each positive integer n we can find vn, v′n∈ U and kn∈ K with
vn6= vn′, d(vn, C) < 1
n, d(v′n, C) < 1
n and ϕ(kn, vn) = ϕ(kn, vn′).
By the compactness of C and K, we can extract subsequences vni, v′ni and kni which converge respectively to v∞, v′∞ ∈ C and k∞∈ K. By continuity of ϕ, we see that ϕ(k∞, v∞) = ϕ(k∞, v′∞).
From (ii), it results that v∞= v′∞. Since vn6= v′n, we can set un=
vn−vn′
kvn−vn′k. Extracting a subsequence if necessary, we can suppose that uni → u∞. This limit u∞ is also of norm 1. As vni and v′ni both converge to v∞ = v∞′ , for i large, the segment between vni
and v′ni is contained in the open set U . Hence for i big enough we can write
0 = ϕ(kni, vni) − ϕ(kni, vn′i)
= Z 1
0
∂ϕ
∂v(kni, svni+ (1 − s)v′ni)(vni− vn′i) ds,
dividing by kvn′i− vnik and taking the limit as ni→ ∞, we obtain 0 =
Z 1 0
∂ϕ
∂v(k∞, v∞)(u∞) ds
= ∂ϕ
∂v(k∞, v∞)(u∞).
However ∂ϕ∂v(k∞, v∞) is an isomorphism, since (k∞, v∞) ∈ K × C.
But ku∞k = 1, this a contradiction. We thus showed the existence of an integer n such that the restriction of ϕk on Vn is injective, for each k ∈ K. The continuity of (k, v) 7→ ∂ϕ∂v(k, v) and the fact that ∂ϕ∂v(k, v) is an isomorphism for each (k, v) in the compact set K ×C, show that, taking n larger if necessary, we can suppose that
∂ϕ
∂v(k, v) is an isomorphism for each (k, v) ∈ K × Vn. The usual inverse function theorem then shows that ϕk restricted to Vn is a local diffeomorphism for each k ∈ K. Since we have already shown that ϕk is injective on Vn, it is a diffeomorphism of Vnon an open subset of Rm.
The following lemma is a simple topological result that de-serves to be better known because it simplifies many arguments.
Lemma 2.7.2. Let X be a topological space, and let Y be a locally compact locally connected Hausdorff space. Suppose that ϕ : X × U → Y is continuous, where U is an open subset of Y , and that, for each x ∈ X, the map ϕx : U → Y, y 7→ ϕ(x, y) is a homeomorphism onto an open subset of Y . Then, the map Φ : X × U → X × Y, (x, y) 7→ (x, ϕ(x, y)) is an open map, i.e. it maps open subsets of X × U to open subsets of X × Y . It is thus a homeomorphism onto an open subset of X × Y .
65 Proof. It is enough to show that if V is open and relatively com-pact in Y , with ¯V ⊂ U , and x0 ∈ X, y0 ∈ Y are such that y0 ∈ ϕx0(V ), then, there exists a neighborhood W of x0 in X and a neighborhood N of y0 in Y , such that ϕx(V ) ⊃ N , for each x ∈ W . In fact, this will show the inclusion W × N ⊂ Φ(W × V ).
As ϕx0(V ) is an open set containing y0, there exists N , a compact and connected neighborhood of y0 in Y , such that N ⊂ ϕx0(V ).
Since ∂V = ¯V \ V is compact and N ∩ ϕx0(∂V ) = ∅, by continuity of ϕ, we can find a neighborhood W of x0 such that
∀x ∈ W, ϕx(∂V ) ∩ N = ∅. (*) We now choose ˜y0 ∈ V , such that ϕx0(˜y0) = y0. Since N is a neighborhood of y0 and ϕ is continuous, taking W smaller if necessary we can assume that
∀x ∈ W, ϕx(˜y0) ∈ N. (**) By condition (∗), for x ∈ W , we have ϕx(V ) ∩ N = ϕx( ¯V ) ∩ N , therefore the intersection ϕx(V ) ∩ N is both open and closed as a subset of the connected space N . This intersection is not empty because it contains ϕx(˜y0) by condition (∗∗). By the connectedness of N , his of course implies that ϕx(V ) ∩ N = N .
Lemma 2.7.3 (Tilting). Let k·k be a norm on Rn. We denote by ˚Bk·k(0, R) (resp. ¯Bk·k(0, R)) the open (resp. closed) ball of Rn of center 0 and radius R for this norm. We suppose that K is a compact space and that ǫ, η, C1 and C2 are fixed > 0 numbers, with C1> C2.
Let θ : K×] − ǫ, ǫ[× ˚Bk·k(0, C1+ η) → Rn be continuous map such that
(1) for each fixed k ∈ K, the map (t, v) 7→ θ(k, t, v) has every-where a partial derivative ∂θ∂t, and this partial derivative is itself C1;
(2) the map θ and its partial derivatives ∂θ∂t,∂∂t22θ,∂v∂t∂2θ = ∂v∂ ∂θ
∂t
are continuous on the product space K×]−ǫ, ǫ[× ˚Bk·k(0, C1+ η);
(3) for each k ∈ K and each v ∈ ¯Bk·k(0, C1+ η), we have
∂θ
∂t(k, 0, v) = v;
(4) for each (k, v) ∈ K × B(0, C1+ η), we have θ(k, 0, v) = θ(k, 0, 0).
Then, there exists δ > 0 such that, for each t ∈ [−δ, 0[∪]0, δ]
and each k ∈ K, the map v 7→ θ(k, t, v) is a diffeomorphism of B˚k·k(0, C1+ η/2) onto an open subset of Rn, and moreover
{θ(k, t, v) | v ∈ ¯Bk·k(0, C1)} ⊃ {x ∈ Rn| kx − θ(k, 0, 0)k ≤ C2|t|}.
Proof. Let us consider the map
Θ(k, t, v) = θ(k, t, v) − θ(k, 0, v)
t ,
defined on K × ([−ǫ, 0[∪]0, ǫ]) × ˚Bk·k(0, C1+ η). We can extend it by continuity at t = 0 because
Θ(k, t, v) = Z 1
0
∂θ
∂t(k, st, v) ds (∗) The right-hand side is obviously well-defined for t = 0, and equal to ∂θ/∂t(k, 0v) = v. Moreover, upon inspection of the right-hand side of (∗), the extension Θ : K×] − ǫ, ǫ[× ˚Bk·k(0, C1+ η) → Rn is such that for each fixed k ∈ K, the map (t, v) → Θ(k, t, v) is C1, with
∂Θ
∂t(k, t, v) = Z 1
0
∂2θ
∂t2(k, st, v)s ds,
∂Θ
∂v(k, t, v) = Z 1
0
∂2θ
∂v∂t(k, st, v) ds.
Therefore both the partial derivatives ∂Θ/∂t, ∂Θ/∂v are continu-ous on the product space K×] − ǫ, ǫ[× ˚Bk·k(0, C1+ η). Let us then define the map ˜Θ : K×] − ǫ, ǫ[× ˚Bk·k(0, C1+ η) → R × Rn by
Θ(k, t, v) = (t, Θ(k, t, v)).˜
67 To simplify, we will use the notation x = (t, v) to indicate the point (t, v) ∈ R × Rn= Rn+1. The map ˜Θ is obviously continuous, and the derivative ∂ ˜Θ∂x is also continuous on the product space K×] − ǫ, ǫ[× ˚Bk·k(0, C1+ η). Since Θ(k, 0, v) = v, for each (k, v) ∈ K × ˚Bk·k(0, C1+ η), we find that
∂ ˜Θ
∂x(k, 0, v) =
1 0
∂Θ
∂t(k, 0, v) IdRn
,
where we used a block matrix to describe a linear map from the product R × Rninto itself. It follows that ∂ ˜∂xΘ(k, 0, v) is an isomor-phism for each (k, v) ∈ K × ˚Bk·k(0, C1+ η).
Since K and ¯Bk·k(0, C1+ η/2) are compact, using the inverse function theorem 2.7.1, we can find δ1 > 0 and η′ ∈]η/2, η[ such that, for each k ∈ K, the map (t, v) 7→ Θ(k, t, v) is a C1 diffeomor-phism from the open set ] − δ1, δ1[× ˚Bk·k(0, C1+ η′) onto an open set R × Rn. It follows that, for each (k, t) ∈ K×] − δ1, δ1[, the map v 7→ ˜Θ(k, t, v) is a C1 diffeomorphism ˚Bk·k(0, C1+ η/2) onto some open subset of Rn. By lemma 2.7.2, we obtain that the image of K×] − δ1, δ1[× ˚Bk·k(0, C1) by the map (k, s, v) 7→ (k, ˜Θ(k, s, v)) is an open subset of K × R × Rn. This open subset contains the compact subset K × {0} × ¯Bk·k(0, C2), since ˜Θ(k, 0, v) = (0, v).
We conclude that there exists δ > 0 such that the image of K×] − δ1, δ1[× ˚Bk·k(0, C1) by map the (k, s, v) 7→ (k, ˜Θ(k, s, v)) contains K × [−δ, δ] × ¯Bk·k(0, C2). Hence, for (k, t) ∈ K × [−δ, δ], the image of ˚Bk·k(0, C1) by the map v 7→ Θ(k, t, v) contains ¯Bk·k(0, C2).
Since we have
θ(k, s, v) = sΘ(k, s, v) + θ(k, 0, v) θ(k, 0, v) = θ(k, 0, 0),
we can translate the results obtained for Θ in terms of θ. This gives that, for s 6= 0, and |s| ≤ δ, the map v 7→ θ(k, s, v) is also a diffeomorphism of ˚Bk·k(0, C1 + η/2) on an open subset of Rn and that the image of ¯Bk·k(0, C1) by this map contains the ball B(θ(k, 0, 0), C¯ 2s).
Theorem 2.7.4 (Existence of local extremal curves). Let L : T M → M be a non-degenerate Cr Lagrangian, with r ≥ 2. We
fix a Riemannian metric g on M . For x ∈ M , we denote by k·kx the norm induced on TxM by g. We call d the distance on M associated with g.
If K ⊂ M is compact and C ∈ [0, +∞[, then, there exists ǫ > 0 such that for x ∈ K, and t ∈ [−ǫ, 0[∪]0, ǫ], the map π ◦ φLt is defined, and induces a diffeomorphism from an open neighborhood of {v ∈ TxM | kvkx ≤ C} onto an open subset of M . Moreover, we have
π ◦ φLt({v ∈ TxM | kvkx ≤ C}) ⊃ {y ∈ M | d(y, x) ≤ C|t|/2}.
To prove the theorem, it is enough to show that for each x0 ∈ M , there exists a compact neighborhood K, such that the conclusion of the theorem is true for this compact neighborhood K. For such a local result we can assume that M = U is an open subset of Rn, with x0 ∈ U . In the sequel, we identify the tangent space T U with U × Rn and for x ∈ U , we identify TxU = {x} × Rn with Rn. We provide U × Rn with the natu-ral coordinates (x, v) = (x1, · · · , xn, v1, · · · , vn). We start with a lemma which makes it possible to replace the norm obtained from the Riemannian metric by a constant norm on Rn.
Lemma 2.7.5 (Distance Estimates). For each α > 0, there exists an open neighborhood V of x0with ¯V compact ⊂ U and such that
(1) for each v ∈ TxU ∼= Rn and each x ∈ ¯V we have (1 − α)kvkx0 ≤ kvkx≤ (1 + α)kvkx0; (2) for each (x, x′) ∈ ¯V we have
(1 − α)kx − x′kx0 ≤ d(x, x′) ≤ (1 + α)kx − x′kx0. Proof. For (1), we observe that, for x → x0, the norm kvkx tends uniformly to 1 on the compact set {v | kvkx0 = 1}, by continuity of the Riemannian metric. Therefore for x near to x0, we have
∀v ∈ Rn, (1 − α) <
v kvkx0
x
< (1 + α).
For (2), we use the exponential map expx: TxU → U , induced by the Riemannian metric. It is known that the map exp : T U =
69 U ×Rn→ U ×U, (x, v) 7→ (x, expxv) is a local diffeomorphism on a neighborhood of (x0, 0), that expx(0) = x, and D[expx](0) = IdRn. Thus, there is a compact neighborhood ¯W of x0, such that any pair (x, x′) ∈ ¯W × ¯W is of the form (x, expx[v(x, x′)]) with v(x, x′) → 0 if x and x′ both tend to x0. The map (x, v) 7→ expx(v) is C1, therefore, using again expx(0) = x, D[expx](0) = IdRn, we must have
expxv = x + v + kvkx0k(x, v),
with limv→0k(x, v) = 0, uniformly in x ∈ ¯W . Since d(x, expxv) = kvkx, for v small, it follows that for x, x′ close to x0
kx − x′kx
d(x, x′) = kv(x, x′)+kv(x, x′)kx0k(x, v(x, x′))kx kv(x, x′)kx .
We can therefore conclude that kx−xd(x,x′k′)x → 1, when x, x′ → x0. But we also have kx−xkx−x′′kkx0x → 1 when x → x0, we conclude that
d(x,x′)
kx−x′kx0 is close to 1, if x and x′ are both in a small compact neighborhood of x0.
Proof of the theorem 2.7.4. Let us give α and η two > 0 numbers, with α enough small to have
C
1 − α < C 1 + α+η
2 1
2(1 − α) < 1 1 + α.
We set C1 = C/(1 + α). Let ¯W ⊂ U be a compact neighborhood of x0. Since ¯W × ¯Bk·kx0(0, C1 + η) is compact, there exists ǫ > 0 such that φLt is defined on ¯W × ¯Bk·kx0(0, C1+ η) for t ∈] − ǫ, ǫ[.
We then set θ(x, t, v) = π ◦ φLt(x, v). The map θ is well defined on W × [−ǫ, ǫ] × ¯¯ Bk·kx0(0, C1+ η). Moreover, since t 7→ φLt(x, v) is the speed curve of its projection t 7→ θ(t, x, v), we have
φLt(x, v) = (θ(x, t, v),∂θ
∂t(x, t, v)), and
θ(x, 0, v) = x, and ∂θ
∂t(x, 0, v) = v.
Since the flow φLt is of class Cr−1, see theorem 2.6.5, both maps θ and ∂θ∂t are of class Cr−1, with respect to all variables. Since r ≥ 2, we can then apply the tilting lemma 2.7.3, with C1 = C/(1 + α) and C2 = C/2(1 − α), to find δ > 0 such that, for each x ∈ ¯W , and each t ∈ [−δ, 0[∪]0, δ], the map (x, v) 7→ π ◦ φLt(x, v) induces a C1 diffeomorphism from {v | kvkx0 < η/2 + C/(1 + α)} onto an open subset of Rnwith
{π ◦ φLt(x, v) | kvkx0 ≤ C
1 + α} ⊃ {y ∈ Rn| ky − xkx0 ≤ Ct 2(1 − α)}.
Since π ◦ φL0(x, v) = x, taking ¯W and δ > 0 smaller if necessary, we can assume that ¯W ⊂ V and
{πφLt(x, v) | t ∈ [−δ, δ], x ∈ ¯W , v ∈ ¯B(0, C/(1 + α) + η)} ⊂ V, where V is given by lemma 2.7.5. Since ¯W ⊂ V , by what we obtained in lemma 2.7.5, for x ∈ ¯W , for every R ≥ 0, we have
{v ∈ TxU | kvkx0 ≤ R
1 + α} ⊂ {v ∈ TxU | kvkx ≤ R}
⊂ {v ∈ TxU | kvkx0 ≤ R 1 − α}.
As ¯W is compact and contained in the open set V , for t > 0 small and x ∈ ¯W , we have
{y ∈ M | d(y, x) ≤ Ct 2 } ⊂ V, hence again by lemma 2.7.5
{y ∈ M | d(y, x) ≤ Ct
2 } ⊂ {y ∈ V | ky − xkx0 ≤ Ct 2(1 − α)}.
Therefore by the choices made, taking δ > 0 smaller if necessary, for t ∈ [−δ, δ], and x ∈ ¯W , the map π◦φLt is a diffeomorphism from a neighborhood of {v ∈ TxU | kvkx ≤ C} onto an open subset of U such that
π ◦ φLt({v ∈ TxU | kvkx ≤ C}) ⊃ {y ∈ V | d(y, x) ≤ C|t|
2 }.
71