数理解析研究所講究録
第 1542 巻 2007 年 12-23

12

AN APPLICATION OF AUBRY-MATHER THEORY IN
LORENTZIAN GEOMETRYANTONIO SICONOLFI

1. OVERVIEW
The results presented in this note have been obtained in the frameworkof a research in progress with Albert Fathi, see [4].
We give an application of the Aubry-Mather theory for non regular
Hamiltonian, recently developed in [6], see also [5], [3] in Lorentzian geometry. Namely, we provide an alternative proof, based on Aubry-Mather
theory, of the existence of a smooth time function for Lorentzian manifolds
enjoying suitable assumptions. Our approach is related to that of [9], where
Aubry-Mather theory is exploited to construct a Lyapunov function for a
multivalued dynamics.
We make in this way a connection which is, at a first sight, quite surprising since the Aubry-Mather theory concerns the qualitative analysis ofHamilton-Jacobi equations at the critical value, i.e. when subsolutions, but
not strect subsolutions, do exist, while, in the time function problem, no
Hamiltonians are involved. In what follows we try to explain how to bridge
this gap.

2. AUBRY SETIn terms of Hamilton-Jacobi equations the issue we are interested on canbe described as follows:
Given an Hamiltonian $H(x,p)$ defined on a cotangent bundle of a (smooth,$bo$ undaryless, connected, paracompact, Hausdorff) manifold $M$ , we look fora smoothstrict subsolution of the equation$(C^{\infty})$$H(x, Du)=0$ ,(1)or at least for a strict subsolution which is smooth in a open subset of $M$ aslarge as possible. Here and throughout the paper subsolution means locally

Lipschitz-continuous $a.e$ . subsolution. The Hamiltonian $H$ is required to be13continuous in both arguments, convex and coercive in . Clearly, to makethe problem sensible, we also have to assume that some subsolutions to (1)do exist.
We then consider the point $sy$ of $M$ possessing a neighborhood wheresome subsolution to (1) satisfies
$p$

$u$

$H(x, Du(x).)\leq-\delta$

and denote by
We have:$M_{0}$

a.e. for some$\delta>0$(2),

the (possibly empty) open subset made up by such points.
$-$

Theorem 2.1. There exists a subsolution to (1), which is smooth and strictin$M_{0}$.

The proof of the Theorem can be found in [5], [6]. It is basically structured in two steps. First, by exploiting the convex character of $H$ , it isconstructed, through an infinite convex combination, a strict subsolution in. This function is then regularized, in order to satisfy the statement, byusing a partition of unity technique and a mollification procedure.
The bad points are then those around which no subsolution satisfies(2). This (possibly empty) set is named after Aubry. An expert of the field
knows that it can be actually nonempty only if the value is critical for $H$ .
We skip here this kind of results since we do not need them in what follows.
It immediately comes from Theorem 2.1:
$M_{0}$$u$$0$

Theorem 2.2. There exists a strict smooth subsolution to (1) if and onlyis empty.
if the Aubry set$M\backslash M_{0}$It is crucial for the subsequent analysis to provide a metric characterization of the Aubry set. We consider the set-valued covector field$xarrow*Z_{x}=\{p :H(x,p)\leq 0\}\subset T_{M}^{*}(x)$,indicates the cotangent space to $M$ at . According to theassumptions made on $H$ , we see that is compact convex valued and continuous with respect to the Fell topology, at least at the points where theis nonempty, which is actually the case we will be interested
interior ofof aon in what follows. We recall that a sequence of closed subsets$K$in the Fell sense provided
topological space converges to a closed subset
that

where

$x$

$T_{M}^{*}(x)$

$Z$

$x$

$Z_{x}$$K_{n}$

for any subsequence
for any $p\in K$$n_{i}$

and $p_{n:}arrow p\Rightarrow p\in K$.
withthere exists$p_{n_{i}}\in K_{n_{i}}$$p_{n}\in K_{n}$$p_{n}arrow p$

14

We also consider the support function$(x, q) rightarrow\sigma_{Z}(x, q)=\max\{(p, q\rangle_{x} :p\in Z_{x}\}$,where is a tangent vector to $M$ at $x(q\in T_{M}(x))$ , anddenotes the$T_{M}^{*}(x)\cross
T_{M}(x)$pairing in. We see thatis continuous in both variables
and convex positively homogeneous in . We define for any continuous
piecewise
curve defined on some interval $[a, b]$ the intrinsic length
via the formula$q$$\langle\cdot, \cdot\rangle_{x}$$\sigma_{Z}$$q$$C^{1}$$\xi$$\ell_{Z}$$\ell_{Z}(\xi)=\int_{a}^{b}\sigma_{Z}.(\xi,\dot{\xi})dt$,(3)

note that, thanks to positive homogeneity of , the value of this integral
is not affected by orientation-preserving change of parameter. We makeprecise that any curve considered in the remainder of the paper will be.
intended to be continuous piecewise
It is not difficult to prove, see [6], that there are subsolutions to (1) if andonly if the intrinsic length of any cycle (i.e. closed curve) is nonnegative. In
the following crucial characterization, see [6], we use the fact that the manifold $M$ , being paracompact, can be endowed with a (complete) Riemannian$\sigma_{Z}$$C^{1}$

metric.

Theorem 2.3. Let be any Riemannian metr\’ic on $M$ , and denote by
the associated length functional defined on the curves of $M$ .
A point belongs to the Aubry set
if and only if there $exist_{8}a$sequenceof cycles based on such that$\ell^{\tilde{g}}$$\tilde{g}$$M\backslash M_{0}$

$y$

$y$

$\xi_{n}$$\lim_{n}\ell_{Z}(\xi_{n})=0$

and$\inf_{n}\ell^{\tilde{g}}(\xi_{n})>0$.

An important remark is now in order. In the previous analysis we havenever make direct use of the Hamiltonian $H$ , all the notions introduced being
instead based on the set-valued map $Z$ . In particular the property of being
subsolution to (1) for a Lipschitz-continuous function can be equivalently
expressed by requiring to solve the partial differential inclusion$Du(x)\in Z_{x}$a.e. in M.
(4)
$u$

$u$

Similarly a smooth function$\varphi$

is a strict subsolution of (1) if and only if$D\varphi(x)\in Z_{x}^{\circ}$

for $x\in M$ ,whereis the interior of . Looking at the issue from this angle, we
furthermore understand that the assumption of compactness on the images$z_{x}^{o}$$Z_{x}$

15

of $Z$ can be eliminated. The possibility of finding smooth functions withdifferential contained in the interior of becomes in fact greater the larger
is the size of .
To make more precise this intuition, we consider, without reference to any
initial Hamiltonian, a continuous closed convex valued covector field $Z$ , andwith the closed Riemanniana Riemannian metric on $M$ , then we cut.
cotangent unitary balls having center at , denoted by
We define in this way a convex compact valued map$Z_{x}$$Z_{x}$$Z_{x}$$g$$B_{x}^{g}(0,1)\subset T_{M}^{*}(x)$$0$for $x\in M$ ,$xrightarrow Z_{x}\cap B_{x}^{g}(0,1)$(5)which is still continuous, provided its values possess nonempty interior for
any . Note that (5) depends on the Riemannian metric , by varying it weaccordingly modify the set-valued map as well as the associated intrinsic
length and Aubry set. We can, in conclusion, rephrase part of the previous
results and state them in the more convenient form for later use as follows:
$x$

$g$

Theorem 2.4. Let $Z$ be a continuous (in the Fell sense) closed convexvalued covector field of M. If there is a Riemannian metric on $M$ suchthat.
has nonempty interior for any
admits a solution,
ii. the inclusion (4) within place ofiii. the Aubry set associated tois empty,
then there exists a smooth function with
$g$

$a$

$i$

$Z_{x}\cap B_{x}^{g}(0,1)$$x_{f}$$Z_{x}\cap B_{x}^{9}(0,1)$$Z_{x}$$Z_{x}\cap B_{x}^{g}(0,1)$$\varphi$$D\varphi(x)\in z_{x}^{o}$for any .
$x$Remark 2.5. Taking into account the Theorem 2.3, we note that, given ,to prove the emptiness of the associated Aubry set it is enough to exhibita Riemannian metric such that any sequence of cycles with infinitesi) also have infinitesimal
mal intrinsic length (associated to-Riemannian length. We will use this principle in Section 5.
play a quite different role in this context:
The Riemannian metrics, while is involved in the fact theprovides the cutting balls
as sociated length functional appears in the characterization of the Aubry
set given by Theorem 2.3.
$g$$\tilde{g}$$Z_{x}\cap B_{x}^{g}(0,1)$$\tilde{g}$$g,\tilde{g}$$g$$B_{x}^{g}(0,1)$$\tilde{g}$16

3. LORENTZIAN SETTING
We recall some basic material of Lorentzian geometry, following [8], [1].
Unfortunately, to explain our setting, we are forced to introduce not few
definitions, however the terminology is nice, and a gross understanding ofunderlying physical model is not difficult.
As a starting point, we consider the classical 4-dimensional Minkowski
manifold , where the general point $(x, y, z, t)$ is interpreted as $represent_{1}ng$
an event (the first three components are spatial and the last is relative tothe time), together with the indefinite nondegenerate symmetric bilinear
form given by the diagonal matrix diag $(1, 1, 1, -1)$ , mathematically:$g(x, y, z, t)=x^{2}+y^{2}+z^{2}-t^{2}$for any (x,y,z,t).
The crucial role of such form is to select the physically admissible displacements. We think the time as running at the light speed and, in accord withthe relativity theory which predicts that no physical particle can move aster$v=(v_{x}, v_{y}, v_{z}, v_{t})$than light, we qualify as admissible a displacement vectorifor equivalently $g(v)\leq 0$ .
1$\mathbb{R}^{4}$$g$$v_{t}|\geq\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}$We call causal the vectors satisfying the previous inequality; among them wefurther distinguish the timelike and lightlike vectors at which is strictly
negative and vanishing, respectively. We refer to the remaining vectors(with strictly positive) as spacelike.
4 asThis terminology applies to curves and smooth hypersufaces ofwell, looking at the tangent vectors. A timelike curve is then the path of anobserver traveling less than the speed of light, a spacelike curve can have ageometric interest but does not describe a physically admissible motion.
The causal vectors make up a closed nonconvex cone with vertex at .
by choosing one of the two convex
We determine a time-orientation ofcones composing it, say the one with nonnegative time component. This iscalled future directed causal cone. It is salient in the sense that it does notcontain any pair of opposite nonzero vectors.
$M$
More generally a future-oriented Lorentzian manifold , or in short a
space-time, of dimension $N$ is a manifold with the continuous assignment toeach tangent space of an indefinite nondegenerate symmetric bilinear form
with signature$(+, +, \cdots+, -)$ ,
$g$

$g$

$\mathbb{R}$$0$$R^{4}$

17

i.e. with all the eigenvalues positive, except one, which is negative, and thecontinuous choice of a future-directed causal cone, that will be denoted,
from now on, by $C_{x}\subset T_{M}(x)$ . We will be particulary interested in what
, which isfollows by the alient closed cone-valued vector field ト
continuous in the Pell topology. Note that any tangent space to $M$ is an$N$-dimensional Minkowski manifold, up to the choice of a basis.
We refer to any curve satisfying the inclusion $\xi(t)\in C_{\xi(t})$ (resp. $\xi(t)\in$
$x$

$s$

$arrow C_{x}$$\xi$for a.e. , as future-directed causal (resp. timelike). Given in $M$ ,we denote by $J^{+}(x)$ (resp. $J^{-}(x)$ ) the set of point$sy$ contained in somefuture-directed causal curve starting at (resp. ending at ).
$C_{\xi(t)})\circ$

$x$

$t$

$x$

$x$

4. TIME FUNCTIONS
A classical subject of investigation in General Relativity is to look forsurfaces of simultaneity in a general space-time , where time and spacevariables are not separated any more, playing the same role as the levels$t=$ cmstant in a Minkowski space.
The sets we are interested on should at least be achronal, in the sensethat no points of them can be connected by future-directed timelike curves.
One should also more stringently ask that any inextendible timelike curveintersects it exactly once, where a curve is qualified as inextendible if thelimits at the endpoints of its interval of definition do not exist. A surfaceof this type is termed after Cauchy, since it is the natural region to pose
initial conditions for hyperbolic equations, as Einstein’s one.
These issues are strictly related to the existence of time functions, i.e.
functions strictly increasing along each future-directed causal curve, being
any level set of such a function achronal. If, in addition, the time function
runs $from-\infty to+\infty$ along any inextendible future-directed causal curve,
then every level set is also a Cauchy surface.
Quite understandingly, it turns out that suitable assumptions on thecausal structure of the space-time are needed for the existence of suchobjects. The basic one is called weakly causality, and requires no timenot tolike curves to be a cycle, or equivalently the inclusion
have closed curves as solutions. The violation of it would apparently yieldcausality breakdowns, in that one could travel into one’s past. Note that
this condition rules out compact space-times since it can be proved (see [1],Proposition 3.10) that compactness and weak causality are incompatible.
$M$$\xi(t)\in C_{\xi(t)}\circ$18

The previous notion can be strengthen by requiring some stability withrespect to small perturbations of the Lorentzian structure. We first define
the order $relation\succ$ in the family of convex cones with vertex contained, as follows:
in $T_{M}(x)$ , for
$0$

some.

$x$

$D_{0}\succ C_{0}$if$D_{0}^{\circ}\supset C_{0}\backslash \{0\}$.

We say that $M$ is stably causal if there is a continuous closed cone valued
for any , such that the inclusion
covector field $D$ , withis not solved by any closed curve. If, in addition, the set $J^{+}(x)\cap J^{-}(y)$ is$M$ will be called globally hyperbolic. The role playedcompact for any $x,$
by cycles in the definition of (weak, stable) causality casts some light on thepossible connection between the topic we are treating here and the results
exposed in Section 2. Recall, in fact, that the definition of Aubry set has
been also given in term of cycles, see Theorem 2.3.
The stable causality and the global hyperbolicity result to be equivalent
to the existence of a time function and of a Cauchy surface, respectively, see[8], [1]. However these results are, in their classical formulation, of topological nature. More precisely Geroch establishes in his seminal paper [7] theequivalence for an $N$-dimensional $space-timeM$ between global hyperbolicity and the existence of a topological Cauchy $(N-1)$ -dimensional surface,i.e. a topological subspace of $M$ where any point possess a neighborhood. The derived splitting Theorem
homeomorphic to an open subset of$M$, where is a
is homeomorphic tostates that a global hyperbolic$M$
topological hypersurface of . A modification of Geroch’s technique allows
to show the already announced equivalence between stable causality andthe existence of a continuous time function.
The possibility of smoothing up these results has remained as an open
question for a long time, and, after some fallacious attempts, it has been
only recently solved by Bernal and S\’anchez in [2]. They assume the global
hyperbolicity of thc spacc-timc, and starting from Geroch’s splitting theorem, which is in turn based on suitable probability measures defined on $M$ ,they prove the existence of a smooth Cauchy surface, and from this they
construct a smooth time function whose level sets are Cauchy surfaces.
We look at the subject from a new angle and propose a new deduction ofthese results by using Aubry-Mather theory.
$D_{x}\succ C_{x}$$\xi(t)\in D_{\xi(t)}\circ$

$x$

$y,$

$\mathbb{R}^{N-1}$$\mathbb{R}\cross S$

$S$

19

5. MAIN RESULT
To keep our treatment as light as possible, we restrict ourselves to explain
how Theorem 2.4 can be applied to prove:

Theorem 5.1. Any stably causal space-time $M$ admits a smooth time function.
We sacrifice precision for the sake of clarity and brevity. We consider thedual cone$C_{x}^{*}=${$p\in T_{M}^{*}(x)$:$\langle p,$

for any $q\in C_{x}$ }.

$q\rangle_{x}\leq 0$

The time function evoked in the statement of the theorem is nothing but asmooth solution of the partial differential inclusion$-Du(x)\in C_{x}^{*}\circ$.

(6)

We therefore proceed to check if the assumptions of Theorem 2.4 are satisfied. We see that it inherits from $C$ the property
by the set-valued map
is nonempty
of being continuous in the Fell sense, moreover the interior offor every , sincedoes not contain any linear subspace. This implies thatthe interior ofis also nonempty for every Riemannian metric$M$
on . Finally the null function is a solution of the partial differential
in place of . In order to get existence
inclusion (4) withof solutions to (6), it is thus left to prove the emptiness of the associatedAubry set for a suitable choice of .
Given a Riemannian metric , we define an intrinsic length for curves,
through the integral formula (3) with the support function
denoted byofas integrand, and look at the intrinsic length of thecycles, in accord with Theorem 2.3. Note that the intrinsic length of any
curve is nonnegative since $0\in C_{x}$ for any , in addition$xrightarrow C_{x}^{*}$$C_{x}^{*}$$C_{x}$$x$$C_{x}^{*}\cap B_{x}^{g}(0,1)$$g$$C_{x}^{*}\cap B_{x}^{g}(0,1)$$Z_{x}$$g$$g$$p_{C}^{g}$$\sigma_{C}^{g}$$C_{x}^{*}\cap B_{x}^{g}(0,1)$$x$$\sigma_{C}^{g}(x, q)=0$

for all $x\in M,$ $q\in C_{x}$ ,so that the intrinsic length of any causal curve is zero, and consequently,

in the light of Theorem 2.3, any causal nonconstant cycle is contained in
the Aubry set, regardless to the choice of . The stable causality rules out
the possibility of having such cycles. We are going furthermore to provethat this condition allows a choice of yielding an empty Aubry set, which
implies Theorem 5.1. For this it will be enough to show the following:
$g$

$g$

20

Proposition 5.2. There exist two Riemannian metrics
for every nonconstant curve : $[a, b]arrow M$ with

$g$

and

$\tilde{g}$such that$\xi$$l_{C}^{g}(\xi)<mIn\{\ell^{\tilde{g}}(\xi), 1/2\}$

we can find a curve 7, with of$\dot{\gamma}(t)\in D_{\gamma(t);}^{\circ}$(7),

joining$\xi(a)$to$\xi(b)$.

In the statement $D$ denotes the cone valued map, withfor any, satisfying the causality condition. We indeed deduce from the stablecausality that $\xi(a)\neq\xi(b)$ , and so any satisfying (7) cannot be a cycle.
Consequently every sequence of cycles with infinitesimal intrinsic length ,must also have infinitesimal Riemannian length . This shows that theis empty, thanks to Remark 2.5.
Aubry set corresponding tofor any implies that the values of such cone maps
The relation
are locally separated by pairs of constant cones. More precisely it can beproved that there exists a countable atlas of smooth charts$D_{x}\succ C_{x}$

$x$

$\xi$

$\ell_{C}^{g}$$\ell^{\overline{9}}$$p_{C}^{g}$$D_{x}\succ C_{x}$$x$$\varphi_{n}$and two sequences of cones:$U_{n}\subset Marrow V_{n}\subset \mathbb{R}^{N}$$C_{0}^{n},$$D_{0}^{\mathfrak{n}}$ofsuch that$\mathbb{R}^{N}$$D\varphi_{n}(\varphi_{n}^{-1}(y))D_{\varphi_{\overline{n}^{1}}(y)}\succ D_{0}^{n}\succ C_{0}^{n}\succ D\varphi_{n}(\varphi_{n}^{-1}(y))C_{\varphi_{\overline{n}^{1}}(y)}$,.
for each , andA crucial step is then to establish a simple property linking the order rela$tion\succ on$ the family of cones, natural and intrinsic length, in the Euclidean$B(O, 1)$ ,spaoe
for a pair of constant cone maps. The symbols
will denote Euclidean norm, length of curves, unitary dual closed ball
, respectively.
and pairing in$n$$y\in V_{n}\subset \mathbb{R}^{N}$$\mathbb{R}^{N}$$|\cdot|,$$\ell,$$\langle\cdot, \cdot\rangle$$(\mathbb{R}^{N})^{*}\cross \mathbb{R}^{N}$

,
and
with
Lemma 5.3. Given two convex conesof$C_{0}\cap
B(0,1)$
lengthsupportand thethe
we indicate byfunction ofrespectively.
(3)
integrand,
with
as
by
defined fomula
There exists a positive such that for any curve $de_{O}fined$ in some interval$[a, b]$ , satisfying, one has $\xi(b)-\xi(a)\in D_{0}$ .
$C_{0}$$D_{0}$$\mathbb{R}^{N}$$\sigma_{1}(\cdot)_{f}\ell_{1}$$\sigma_{1}$$\delta$$\xi,$$\ell_{1}(\xi)<\delta\ell(\xi)$Proof. Define$S=\{p\in D_{0}^{*} :Since$D_{0}\succ C_{0}$|p|=1/2\}$.

, there exists $1/4>\delta>0$ such that
$B(p, 2\delta)\subset C_{0}^{*}\cap B(0,1)$

for any $p\in S$ .

$D_{0}\succ C_{0}$21This implies

for any$\sigma_{1}(q)>\langle p, q\rangle+2\delta|q|$$p\in S$$q\in \mathbb{R}^{N},$.

From this we deduce$p_{\iota(\xi)\geq}\langle p, \xi(b)-\xi(a)\rangle+2\delta\ell(\xi)$and we exploit the inequality$\ell_{1}(\xi)<\delta\ell(\xi)$,to get$(p, \xi(b)-\xi(a)\rangle<-\delta P(\xi)<0$,口
which implies the as ertion.
Note that the preceding result can be equivalently expressed by sayingthat the Euclidean segment $\gamma(t)=(1-t)\xi(a)+t\xi(b),$ $t\in[0,1]$ , satisfies$s$for any .
, and, inof $M$ , withWe can determine a compact covercorrespondence, a Riemannian metric such that any point at -Riemannianlies in , for any .
distance less than or equal 1 from a fixed point of
The previous lemma and the local separation property for and $D$ imply:$\dot{\gamma}(t)\in D_{0}^{\circ}$$t$$K_{n}\subset U_{n}$$\{K_{n}\}$$\tilde{g}$$\tilde{g}$$n$$U_{n}$$K_{n}$$C$

Lemma 5.4. For any we can findon an interval $[a, b]$ , with$n$$\delta_{n}>0$$\zeta(a)\in K_{n},$such that if is a curve defined, satisfy ing$\zeta$$\zeta([a, b]\subset U_{n}$$\ell_{C}^{\tilde{g}}(\zeta)<\delta_{n}p_{(\zeta)}\tilde{g}$then we can find a curve , with$\gamma(t)\in D_{\gamma(t)}o$$\gamma$for $a.e$ .

$t_{f}$

joining $\xi(a)$ to$\xi(b)$.

We can construct, by using a smooth partition of unity subordinated toon ,[ with, a smooth function : $Marrow$ ]the cover$M$by setting
for each $n\in$ N. We define a new Riemannian metric on. Therefore
for any $x\in M,$ $p\in T_{M}^{*}(x)$ ,$\{U_{n}\}$$\alpha>1/\delta_{n}$$0,$ $+\infty$$\alpha$$U_{n}$$g$$g=\alpha^{2}\tilde{g}$$|p|_{x}^{9}=1/\alpha(x)|p|_{x}^{\tilde{9}}$where$|\cdot|_{x}^{g}$norm on(resp.

$T_{M}^{*}(x)$) indicates the dual -Riemannian (resp. -Riemannian). This implie,$\tilde{g}$$g$$|\cdot|_{x}^{\tilde{g}}$$q$$\sigma_{C}^{g}(x, q)=\alpha(x)\sigma_{C}^{\tilde{9}}(x, q)$

for any $x\in M,$ $q\in T_{M}(x)$ ,and consequently$\ell_{C}^{9}(\zeta)>1/\delta_{n}p_{c(\zeta)}^{\tilde{g}}$for any$n$and any curve$\zeta$contained in$U_{n}$.

(8)

Sketch of the proof of Proposition 5.2. We choose the previously
introduced Riemannian metrics and . The idea is to find a sequence of$\tilde{g}$

$g$

22

such that the Lemma 5.4 can be applied to$i=1,$ $\cdots j$ , for somedepending on .
We define as the first time such thatis at -Riemannian distance
1 from. We can assume without loss of generality that $\gamma(a)\in K_{1}$ , so.
that
If the -Riemannian distance between
is less than 1 thenand$t_{1}=b$ ; by (8) and the assumption we gettimes $t_{0}=a,$$t_{1},$$\cdots t_{j-1},$ $t_{j}=b$$K_{n}$$\xi|_{[t_{i},t_{i}+1]},$$i$$\xi(t_{1})$$t_{1}$$\tilde{g}$$\xi(a)$$\gamma([a, t_{1}])\subset U_{1}$$\xi(a)$$\tilde{g}$$\frac{1}{\delta_{1}}p_{c(\xi)<\ell_{C}^{g}(\xi)\leq\ell^{\tilde{g}}(\xi)}^{\tilde{g}}$$\xi(t_{1})$.

andare at -Riemannian distance 1 then by (8), theassumption and since the intrinsic length is nonnegative we have

If instead$\xi(a)$$\xi(t_{1}))$$\tilde{g}$$\frac{1}{\delta_{1}}\ell_{C}^{\tilde{9}}(\xi|_{[a,t_{1}]})<\ell_{C}^{9}(\xi|_{[a,t_{1}]})\leq l_{C}^{g}(\xi)\leq 1/2\leq\ell^{\tilde{9}}(\gamma|[a, t_{1}])$.

,, joining
In both cases there is a curve , withtoin view of Lemma 5.4. This completes the proof in the caee where $t_{1}=b$ .
If $t_{1}<b$ , we select a8above atime , and construct acurve , with, joining. We then get, by juxtaposition, a curvetoof the same type connectingto. Through iteration of this procedure we obtain the assertion. Some additional precautions must actually
be taken at the last step, but we skiP this Part of the proof.
$\dot{\gamma}_{1}(t)\in D_{\gamma_{1}(t)}\circ$$\gamma_{1}$$t_{2}$$\dot{\gamma}_{2}(t)\in D_{\gamma_{2}(t)}\circ$$\xi(t_{1})$$\xi(a)$$\xi(t_{1})$$\gamma_{2}$$\xi(t_{2})$$\xi(a)$$\xi(t_{2})$口

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DIP. DI MATEMATICA, UNIVERSIT\‘A DI ROMA (LA $SAPI+NZA$ , P.LE ALDO MoRO
2, 00185 ROMA, ITALY
E-mail address: $si$ conolf Qmat. uniromal. it