Theory and Applications of Categories, Vol. 23, No. 10, 2010, pp. 199–220.

A METRIC TANGENTIAL CALCULUS

We dedicate this article to Dominique Bourn.

ELISABETH BURRONI AND JACQUES PENON

Abstract. The metric jets, introduced here, generalize the jets (at order one) of Charles Ehresmann. In short, for a “good” map f (said to be “tangentiable” at a) between metric spaces, we define its metric jet tangent at a (composed of all the maps which are locally lipschitzian at a and tangent to f at a) called the “tangential” of f at a, and denoted Tf_{a}. So, in this metric context, we define a “new differentiability” (called “tangentiability”) which extends the classical differentiability (while preserving most of its properties) to new maps, traditionally pathologic.

Introduction

First, let us mention that most of the proofs concerning the statements given here can be found in the first chapter of a paper published in arXiv (see [4]).

This paper contains a reference to two previous talks (see [2] and [3]). Our aim being a deep thought inside the fundamentals of differential calculus. Focussing on what is at the heart of the notion of differential, it is the concept of “tangency” which imposed itself in its great simplicity. Now, amazingly, this concept of tangency can be formulated without resorting to the whole traditional structure of normed vector space (here on ℝ), which will be denoted n.v.s.: see Section 1. It is thus the more general structure of metric space in which we work from now on, asking if it is possible to construct a meaningful “metric differential calculus”. We will see that this aim has been essentially reached, even if, on the way, it required the help of an additional structure (the “transmetric” structure).

The first challenge was in the very formulation of a “differential” in a metric context: what can we replace the continuous affine maps with, though they are essential to the definition of the classical differentials? Jets (that we call metric jets in order to emphasize that the metric structure is enough to define them) will play the part of these continuous affine maps, willingly forgetting their algebraic feature. So, we introduce a “new differential” for a map f which admits a tangent at a which is locally lipschitzian at a (such a map being said to be “tangentiable” at a): it is a metric jet, tangent to f at a, called the “tangential” of f at a, and denoted Tf_{a}.

As is well known, jets were first introduced by C.Ehresmann in 1951 [5], in order to adapt Taylor’s expansions to differential geometry; more precisely, his infinitesimal jets (at order one) can be seen as equivalence classes of maps of class C^{1} between differentiable manifolds, under an equivalence relation of tangency. The metric jets we are proposing here are more general (being also equivalence classes of locally lipschitzian maps between metric spaces, under an analogous equivalence relation of tangency).

Received by the editors 2009-04-19 and, in revised form, 2010-02-15.

Published on 2010-03-25 in the Bourn Festschrift.

2000 Mathematics Subject Classification: 58C25, 58C20, 58A20, 54E35, 18D20..

Key words and phrases: differential calculus, jets, metric spaces, categories.

\begin{matrix}c \\ ◯ \end{matrix} Elisabeth Burroni and Jacques Penon, 2010. Permission to copy for private use granted.

In analogy with the distance between two continuous affine maps, we will construct a metric to evaluate the distance between two metric jets: fixing a pair of points (a and a′, respectively in the metric spaces M and M′), the set 𝕁et((M, a), (M′, a′)) of the metric jets from (M, a) to (M′, a′) can, itself, be equipped with a metric structure!

Noticing that our distance between metric jets does not fit to speak of the distance between the tangentials of one map at two different points (these tangentials being metric jets which are tangent to this map at different points), we add a geometrical structure to our metric spaces, inspired by the translations of the usual vector space framework. These particular metric spaces are called “transmetric”. In their frame, we introduce the notion of “free metric jets” (invariant by given “translation jets”). Now, if M and M′ are such transmetric spaces, the set 𝕁et_{free}(M, M′) of the free metric jets from M to M′ can be equipped with a metric structure.

Among these free metric jets, we find the free metric jets of the form tf_{a} “associated” to the tangential Tf_{a} for an f supposed to be tangentiable at a. Finally, when f : M ⟶ M′ is a tangentiable map (i.e. tangentiable at every point of M), we construct its tangential tf : M ⟶ 𝕁et_{free}(M, M′), whose domain and codomain are here metric spaces (since M and M′ are transmetric spaces). We thus obtain a new map on which we can apply the different techniques of the new theory, as, for instance, to study the continuity or the tangentiability of this tangential tf.

For general definitions in category theory (for instance cartesian or enriched categories), see [1]. Acknowledgements: It is a talk about Ehresmann’s jets, given by Francis Borceux at the conference organised in Amiens in 2002 in honour of Andrée and Charles Ehresmann which has initiated our work. Since at that epoch we where interested, in our teaching, in what could be described uniquely with metric tools … hence the idea of the metric jets!

We wish to thank Christian Leruste for his constant help with linguistical matters. We don’t forget our “brother in category” René Guitart: his opening to listening, his interest in our work and his encouragements have been a precious stimulating help.

1. The relation of tangency

The relation of tangency, which is at the heart of the differential calculus, is an essentially metric notion, since it can merely be written: lim_{a≠x→a} \frac{d(f(x),g(x))}{d(x,a)} = 0 for two maps f, g : M ⟶ M′. However, this definition uses the term “lim_{a≠x→a}” which makes sense solely for a point a, not isolated in M; this lead to a more general definition.

Furthermore, to make this relation of tangency compatible with the composition, we have to restrict the type of maps on which we will work. Then we will be able to define the concepts of metric jets and of tangentials.

A METRIC TANGENTIAL CALCULUS  201

The right definition of the notion of tangency will arise from the following equivalences, where M and M′ are two metric spaces, a a fixed point in M and f, g : M ⟶ M′ two maps (a priori without any hypothesis). 1.1. Proposition. The following properties are equivalent :

(i) ∀ε > 0 ∃η > 0 ∀x ∈ M (d(x, a) ≤ η ⟹ d(f(x), g(x)) ≤ εd(x, a)), (ii) f(a) = g(a) and the map C : M ⟶ ℝ_{+} defined by

C(a) = 0 and C(x) = \frac{d(f(x), g(x))}{d(x, a)} ∀x ≠ a

is continuous at a, (iii) there exists a map c : M ⟶ ℝ_{+} which is continuous at a and which verifies:

c(a) = 0 and ∀x ∈ M (d(f(x), g(x)) = c(x)d(x, a)), (iv) there exist a neighborhood V of a in M and a map c : V ⟶ ℝ_{+} which is continuous

at a and which verifies: c(a) = 0 and ∀x ∈ V (d(f(x), g(x)) ≤ c(x)d(x, a)). 1.2. Definition. We say that f and g are tangent at a (which is denoted by f ≻≺_{a} g) if they verify anyone of the equivalent conditions of the above 1.1. 1.3. Remarks.

1. When a is not an isolated point in M (i.e a ∈ \overline{M − \{a\}}), we have: f ≻≺_{a} g iff (f(a) = g(a) and lim_{a≠x→a} \frac{d(f(x),g(x))}{d(x,a)} = 0).

2. When a is an isolated point in M, we have f ≻≺_{a} g for any f and g verifying f(a) = g(a).

3. The relation ≻≺_{a} is an equivalence relation on the set of maps from M to M′ ; this relation ≻≺_{a} is called the relation of tangency at the point a.

4. If f ≻≺_{a} g, then f is continuous at a iff g is continuous at a.

To study the behaviour of the relation of tangency towards composition, we consider the following situation (S) :

\xymatrix{M_{0} \ar[r]^{f_{0}} & M_{1} \ar[r]^{f_{1}} \ar[r]_{g_{1}} & M_{2} \ar[r]^{f_{2}} & M_{3}}

where M_{0}, M_{1}, M_{2}, M_{3} are metric spaces with a_{o} ∈ M_{0}, a_{1} = f_{0}(a_{0}), a_{2} = f_{1}(a_{1}) = g_{1}(a_{1}). Under what conditions do we have one of the implications :

f_{1} ≻≺_{a_{1}} g_{1} ⟹ f_{1}.f_{0} ≻≺_{a_{0}} g_{1}.f_{0} and f_{1} ≻≺_{a_{1}} g_{1} ⟹ f_{2}.f_{1} ≻≺_{a_{1}} f_{2}.g_{1}.

1.4. Remark. The above implications are not true in general, even if the maps are continuous. Consider

M_{0} = M_{1} = M_{2} = M_{3} = ℝ

with

f_{0} = f_{2} : x ↦ x^{1/3} , f_{1} : x ↦ x^{3} and g_{1} : x ↦ 0

Even though f_{1} ≻≺_{0} g_{1}, however f_{1}.f_{0} ⊁≺_{0} g_{1}.f_{0} and f_{2}.f_{1} ⊁≺_{0} f_{2}.g_{1}

In order to give sufficient conditions to make the above implications true (see 1.6 below), we need the following definition:.

1.5. Definition. Let M and M′ be metric spaces, f : M ⟶ M′ a map, and a ∈ M; let also k be a strictly positive real number. We say that :

1. f is locally k-lipschitzian at a (in short k-LL_{a}) if there exists a neighborhood V of a in M for which the restriction f : V ⟶ M′ is k-lipschitzian. f locally lipschitzian at a (in short LL_{a}) means that there exists k > 0 such that f is k-LL_{a}.

2. f is k-semi-lipschitzian at a (in short k-SL_{a}) if we have: ∀x ∈ M (d(f(x), f(a)) ≤ kd(x, a))). f semi-lipschitzian at a (in short SL_{a}) means that there exists k > 0 such that f is k-SL_{a}.

3. f is locally k-semi-lipschitzian at a (in short k-LSL_{a}) if, on a neighborhood V of a in M, the restriction f : V ⟶ M′ is k-SL_{a}. f locally semi-lipschitzian at a (in short LSL_{a}) means that there exists k > 0 such that f is k-LSL_{a}.

Naturally, f is LL or LSL will mean that f is LL_{a} or LSL_{a} at every point a ∈ M.

1.6. Remarks. Let M and M′ be metric spaces, f : M ⟶ M′ a map, and a ∈ M.

1. We have the implications: f LL_{a} ⟹ f LSL_{a} and f LSL_{a} ⟹ f continuous at a. The inverses of these implications are not true (see Section 3).

2. In the above situation (S), let us assume that f_{1} ≻≺_{a_{1}} g_{1} ; we then have the two following implications :

(a) f_{0} LSL_{a_{0}} ⟹ f_{1}.f_{0} ≻≺_{a_{0}} g_{1}.f_{0},

(b) f_{2} LL_{a_{2}} and f_{1}, g_{1} continuous at a_{1} ⟹ f_{2}.f_{1} ≻≺_{a_{1}} f_{2}.g_{1}.

3. Let g : M ⟶ M′ such that f ≻≺_{a} g ; we then have (still by 1.1) the equivalence: f LSL_{a} ⟺ g LSL_{a}.

4. Let E, E′ be two n.v.s., U an open subset of E, a ∈ U and f : U ⟶ E′ a map; let us denote L(E, E′) the set of continuous linear maps from E to E′. We have the implications :

(a) f differentiable at a ⟹ f LSL_{a}, (b) f differentiable and df:U ⟶L(E, E′) continuous at a ⟹ f LL_{a} (in partic-

ular, f of class C^{1} ⟹ f LL).

The inverses of these implications are not true (see Section 3).

5. Let M_{0}, M_{1}, M_{2} be metric spaces ; f_{0} : M_{0} ⟶ M_{1}, f_{1} : M_{1} ⟶ M_{2} two maps, and a_{0} ∈ M_{0}, a_{1} = f_{0}(a_{0}). We have the implications:

(a) f_{0} LSL_{a_{0}} and f_{1} LSL_{a_{1}} ⟹ f_{1}.f_{0} LSL_{a_{0}}.

(b) f_{0} LL_{a_{0}} and f_{1} LL_{a_{1}} ⟹ f_{1}.f_{0} LL_{a_{0}}.

6. Let M_{0}, M_{1}, M_{2} be metric spaces, a_{0} ∈ M_{0}; and also maps f_{0}, g_{0} : M_{0} ⟶ M_{1}, f_{1}, g_{1} : M_{1} ⟶ M_{2}, with a_{1} = f_{0}(a_{0}) = g_{0}(a_{0}). We assume that f_{0} ≻≺_{a_{0}} g_{0} and f_{1} ≻≺_{a_{1}} g_{1} where g_{0} is LL_{a_{0}} and g_{1} is LL_{a_{1}} ; then f_{1}.f_{0} ≻≺_{a_{0}} g_{1}.g_{0}. We could have weakened the hypothesis : g_{0} LSL_{a_{0}} would have been enough.

7. Let M, M_{0}, M_{1} be metric spaces, a ∈ M; and also maps f_{0}, g_{0} : M ⟶ M_{0} and f_{1}, g_{1} : M ⟶ M_{1}. We have the implication:

f_{0} ≻≺_{a} g_{0} and f_{1} ≻≺_{a} g_{1} ⟹ (f_{0}, f_{1}) ≻≺_{a} (g_{0}, g_{1})

8. Let M, M_{0}, M_{1} be metric spaces, a ∈ M; and also maps f_{0} : M ⟶ M_{0} and f_{1} : M ⟶ M_{1}. Then:

(a) f_{0}, f_{1} LSL_{a} ⟹ (f_{0}, f_{1}) LSL_{a},

(b) f_{0}, f_{1} LL_{a} ⟹ (f_{0}, f_{1}) LL_{a}.

This implies that the categories whose objects are metric spaces and whose morphisms are maps which are LSL (resp. LL) at a point, are cartesian categories.

We conclude this section by giving a metric generalization of the mean value theorem (weakening the hypothesis of being differentiable by the one of being LSL).

1.7. Proposition. Let M be a metric space, [a, b] a compact interval of ℝ, k > 0 a fixed real number and f : [a, b] ⟶ M a continuous map which is k-LSL_{x} for all x in the open interval ]a, b[. Then we have d(f(b), f(a)) ≤ k(b − a).

For the proof, we use the following well-known lemma :

1.8. Lemma. Let g : [a, b] ⟶ ℝ be a continuous map and k a real number such that the following property is true:

∀x ∈]a, b[ ∃x′ ∈]a, b] (x′ > x and g(x′) − g(x) ≤ k(x′ − x)).

Then we have g(b) − g(a) ≤ k(b − a).

1.9. Corollary.

1. Let M be a metric space, [a, b] a compact interval of ℝ, F a finite subset of ]a, b[; let also f : [a, b] ⟶ M be a continuous map. Let us assume that, for all x ∈]a, b[−F, the map f is k-LSL_{x} ; then d(f(b), f(a)) ≤ k(b − a).

2. Let E be a n.v.s., U an open subset of E, a, b ∈ U such that [a, b] ⊂ U and F a finite subset of ]a, b[; let also M be a metric space and f : U ⟶ M a continuous map. Let us assume that, for all x ∈]a, b[−F, the map f is k-LSL_{x} ; then, we have again: d(f(b), f(a)) ≤ k∥b − a∥.

2. Metric jets

The metric jets (in short, the jets), which are merely equivalence classes for the relation of tangency, will play the part of the continuous affine maps of the classical differential calculus (but here, without any algebraic properties); it seems natural to equip the set of jets with a metric structure. Thanks to this metric structure, we will enrich the category of jets, between pointed metric spaces, in the category 𝕄et (a well chosen category of metric spaces).

M and M′ being metric spaces, with a ∈ M, a′ ∈ M′, let us denote

𝕃L((M, a), (M′, a′))

the set of maps f : M ⟶ M′ which are LL_{a} and which verify f(a) = a′. These sets 𝕃L((M, a), (M′, a′)) are the “Hom” of a category, denoted 𝕃L, whose objects are pointed metric spaces; this category 𝕃L is a cartesian category (i.e. it has a final object and finite products). Now, since ≻≺_{a} is an equivalence relation on 𝕃L((M, a), (M′, a′)), we set

𝕁et((M, a), (M′, a′)) = 𝕃L((M, a), (M′, a′))/ ≻≺_{a}

2.1. Definition. An element of 𝕁et((M, a), (M′, a′)) is called a jet from (M, a) to (M′, a′).

Let q : 𝕃L((M, a), (M′, a′)) ⟶ 𝕁et((M, a), (M′, a′)) be the canonical surjection. Referring to Section 1, we can compose the jets: q(g.f) = q(g).q(f) when g and f are composable. So, we construct a category, denoted 𝕁et, called the category of jets, whose:

- objects are pointed metric spaces (M, a),

- morphisms φ : (M, a) ⟶ (M′, a′) are jets (i.e. elements of 𝕁et((M, a), (M′, a′))). The previous canonical surjections extend to a functor

q: 𝕃L ⟶ 𝕁et

(constant on the objects) which makes 𝕁et a quotient category of 𝕃L.

2.2. Proposition. The functor q : 𝕃L⟶ 𝕁et creates a cartesian structure on the category 𝕁et (q being constant on the objects, it means that 𝕁et is cartesian and q a strict morphism of cartesian categories).

2.3. Remarks.

1. Let φ : (M, a) ⟶ (M′, a′) be a morphism in 𝕁et and f ∈ φ. If f is locally “antilipschitzian” at a (i.e if there exist k > 0 and a neighborhood V of a on which we have d(f(x), f(y)) ≥ kd(x, y)), then φ is a monomorphism in 𝕁et. The jet of an isometric embedding is thus a monomorphism (in particular, in the case of a metric subspace).

2. Let M be a metric space, V a neighborhood of a ∈ M. Let us set j_{a} = q(j) : (V, a) ⟶ (M, a) where j : V ↪ M is the canonical injection. Then, the jet j_{a} is an isomorphism in 𝕁et.

Time has now come to equip the category 𝕁et((M, a), (M′, a′)) with a metric structure (where (M, a), (M′, a′) ∈ |𝕁et|).

First, we define d(f, g) for f, g ∈ 𝕃L((M, a), (M′, a′)); for such f, g, we consider the map C : M ⟶ ℝ_{+} defined by C(x) = \frac{d(f(x),g(x))}{d(x,a)} if x ≠ a and C(a) = 0. We notice that C is bounded on a neighborhood of a: indeed, since f and g are LL_{a}, there exist a neighborhood V of a and a real number k > 0 such that the restrictions f|_{V} and g|_{V} are k-lipschitzian. Then, for x ∈ V, we have: d(f(x), g(x)) ≤ d(f(x), a′) + d(a′, g(x)) ≤ d(f(x), f(a)) + d(g(a), g(x)) ≤ 2kd(x, a), so that C(x) ≤ 2k for all x ∈ V.

Now, for each r > 0, we set d^{r}(f, g) = sup{C(x) | x ∈ B′(a, r) ∩ V } (where B′(a, r) is a closed ball; this definition does not depend on V for small r). The map r ↦ d^{r}(f, g) is increasing and positive, we can put: d(f, g) = lim_{r→0} d^{r}(f, g) = inf_{r>0} d^{r}(f, g).

2.4. Proposition. Let d : (𝕃L((M, a), (M′, a′)))^{2} ⟶ ℝ_{+} be the map defined just above.

I. For each f, g, h ∈ 𝕃L((M, a), (M′, a′)), the map d verifies the following properties:

1) d(f, g) = d(g, f),

2) d(f, h) ≤ d(f, g) + d(g, h),

3) d(f, g) = 0 ⟺ f ≻≺_{a} g.

II. The map d factors through the quotient, giving a “true” distance on the category 𝕁et((M, a), (M′, a′)), defined by d(q(f), q(g)) = d(f, g) for f, g ∈ LL((M, a), (M′, a′)).

Now, we need to establish some technical properties about what we call the lipschitzian ratio of a jet (that we also need in Section 4).

2.5. Definition. For φ ∈ 𝕁et((M, a), (M′, a′)), we set

ρ(φ) = inf K(φ) , where K(φ) = {k > 0 | ∃f ∈ φ, f is k-LL_{a}}

We call ρ(φ) the lipschitzian ratio of φ. Furthermore, we will say that φ is k-bounded if ρ(φ) ≤ k.

2.6. Proposition. (Properties of ρ)

1. Let (M_{0}, a_{0}), (M_{1}, a_{1}), (M_{2}, a_{2}) ∈ |𝕁et|; and also jets φ_{0} : (M_{0}, a_{0}) ⟶ (M_{1}, a_{1}), φ_{1} : (M_{1}, a_{1}) ⟶ (M_{2}, a_{2}). Then, ρ(φ_{1}.φ_{0}) ≤ ρ(φ_{1})ρ(φ_{0}).

2. For each φ ∈ 𝕁et((M, a), (M′, a′)), we have d(φ, O_{aa′}) ≤ ρ(φ) (where O_{aa′} = q(â′), and â′ : M ⟶ M′ is the constant map on a′).

2.7. Definition. We say that a jet φ is a good jet if the previous inequality given in 2.6.2 becomes an equality.

2.8. Examples. In all that follows (M, a), (M′, a′), (M_{i}, a_{i}) are objects of 𝕁et, i.e pointed metric spaces.

0. We have ρ(O_{aa′}) = 0.

1. For every jet φ : (M, a) ⟶ (M′, a′), where a or a′ are isolated (respectively in M or M′), then ρ(φ) = 0.

2. If we denote π_{i} : (M_{1}, a_{1}) × (M_{2}, a_{2}) ⟶ (M_{i}, a_{i}) the canonical projections in 𝕁et; then d(π_{i}, O_{aa_{i}}) = ρ(π_{i}) = 1 (where a = (a_{1}, a_{2}), with a_{i} non isolated in M_{i}).

3. Let M, M′ be metric spaces, f : M ⟶ M′ an isometric embedding, a a non isolated point in M and a′ = f(a); then d(q(f), O_{aa′}) = ρ(q(f)) = 1. Thus, a being not isolated in M, Id_{(M,a)}, j_{a} and j_{a}^{−1} are good jets (see 2.3 for j_{a}).

2.9. Remarks. Let (M, a), (M′, a′) ∈ 𝕁et.

1. Let us assume that there exists φ ∈ 𝕁et((M, a), (M′a′)) which is an isomorphism in 𝕁et. Then, a is isolated in M iff a′ is isolated in M′.

2. (M, a) is a final object in 𝕁et iff a is isolated in M.

3. Let (M, a), (M_{1}, a_{1}), (M_{2}, a_{2}) ∈ |𝕁et|; consider two jets φ_{1} : (M, a) ⟶ (M_{1}, a_{1}) and φ_{2} : (M, a) ⟶ (M_{2}, a_{2}). Then ρ(φ_{1}, φ_{2}) = sup_{i}(ρ(φ_{i})).

4. For each i ∈ {1, 2}, let (M_{i}, a_{i}), (M′_{i}, a′_{i}) ∈ |𝕁et|, and ψ_{i} : (M_{i}, a_{i}) ⟶ (M′_{i}, a′_{i}) be two jets. Then ρ(ψ_{1} × ψ_{2}) ≤ sup_{i} ρ(ψ_{i}).

2.10. Theorem. Let us consider the following diagram in 𝕁et:

\xymatrix{(M_{0}, a_{0}) \ar[r]^{φ_{0}} \ar[r]_{ψ_{0}} & (M_{1}, a_{1}) \ar[r]^{φ_{1}} \ar[r]_{ψ_{1}} & (M_{2}, a_{2})}

We then have the inequalities:

1. d(ψ_{1}.ψ_{0}, φ_{1}.φ_{0}) ≤ d(ψ_{1}, φ_{1})d(ψ_{0}, O) + ρ(φ_{1})d(ψ_{0}, φ_{0}) (where O = O_{a_{0}a_{1}}: see 2.6).

2. d(ψ_{1}.ψ_{0}, φ_{1}.φ_{0}) ≤ d(ψ_{1}, φ_{1}) + d(ψ_{0}, φ_{0}) if ψ_{0} and φ_{1} are 1-bounded (see 2.5).

\underline{Proof} : 1. Let i ∈ \{0, 1\}; and f_{i} ∈ φ_{i} and g_{i} ∈ ψ_{i}. Then, there exist k_{i}, k′_{i} > 0 and V_{i} a neighborhood of a_{i} in M_{i} such that the restrictions f_{i}|_{V_{i}} and g_{i}|_{V_{i}} are respectively k_{i}-lipschitzian and k′_{i}-lipschitzian.

For all x near a, we have: d(g_{1}.g_{0}(x), f_{1}.f_{0}(x)) ≤ d^{k′_{0}r}(g_{1}, f_{1})d^{r}(g_{0}, â_{1})d(x, a_{0}) + k_{1}d^{r}(g_{0}, f_{0})d(x, a_{0}) (where â_{1} is the constant map on a_{1}), which provides d(g_{1}.g_{0}, f_{1}.f_{0}) ≤ d(g_{1}, f_{1})d(g_{0}, â_{1}) + k_{1}d(g_{0}, f_{0}); hence the wanted inequality.

2. Clear. □

2.11. Remarks.

1. The sets of jets being equipped with their distance, the maps:

𝕁et((M_{0}, a_{0}), (M_{1}, a_{1})) ⟶ 𝕁 et((M_{0}, a_{0}), (M_{2}, a_{2})) : ψ ↦ φ_{1}.ψ and

𝕁et((M_{1}, a_{1}), (M_{2}, a_{2})) ⟶ 𝕁 et((M_{0}, a_{0}), (M_{2}, a_{2})) : ψ ↦ ψ.φ_{0}

are respectively ρ(φ_{1})-lipschitzian and d(φ_{0}, O)-lipschitzian (where φ_{0} and φ_{1} are jets as in 2.10).

2. ψ_{0}, ψ_{1}, O being jets as in 2.10 (with O = O_{a_{0}a_{2}}, O_{a_{1}a_{2}} or O_{a_{0}a_{1}}), we have the inequality: d(ψ_{1}.ψ_{0}, O) ≤ d(ψ_{1}, O)d(ψ_{0}, O).

3. The inequalities obtained just above and in 2.6.1 are both generalisations of the well-known inequality ∥l_{1}.l_{0}∥ ≤ ∥l_{1}∥ ∥l_{0}∥ for composable continuous linear maps (see 2.16 below).

2.12. Corollary.

1. The composition of jets:

𝕁et((M_{0}, a_{0}),(M_{1}, a_{1}))×𝕁et ((M_{1}, a_{1}),(M_{2}, a_{2})) ⟶^{comp} 𝕁et((M_{0}, a_{0}),(M_{2}, a_{2})) is LSL.

2. The category 𝕁et can thus be enriched in the cartesian category 𝕄et (whose objects are the metric spaces and whose morphisms are the locally semi-lipschitzian maps).

3. Let M, M′ be metric spaces, V, V′ be two neighborhoods, respectively of a ∈ M and a′ ∈ M′. Then, the map:

Γ : 𝕁et((V, a), (V′, a′)) ⟶ 𝕁et((M, a), (M′, a′)) : φ ↦ j′_{a′}.φ.j_{a}^{−1}

is an isometry (where j_{a} and j′_{a′} have been defined in 2.3).

2.13. Proposition. (M, a), (M_{0}, a_{0}), (M_{1}, a_{1}) being objects in the category 𝕁et, the following canonical map can is an isometry:

𝕁et((M, a), (M_{0}, a_{0}) × (M_{1}, a_{1}))
can
𝕁et((M, a), (M_{0}, a_{0})) × 𝕁et((M, a), (M_{1}, a_{1}))

2.14. Remark. As isometries are isomorphisms in 𝕄et, it means that 𝕁et is an enriched cartesian category.

We conclude this section with a come back to vectorial considerations.

2.15. Proposition.

1. Let M be a metric space (with a ∈ M) and E a n.v.s. Then, we can canonically equip the sets 𝕃L((M, a), (E, 0)) and 𝕁et((M, a), (E, 0)) with vectorial space structures, making linear the canonical surjection q : 𝕃L((M, a), (E, 0)) ⟶ 𝕁et((M, a), (E, 0)). Besides, the distance on 𝕁et((M, a), (E, 0)) derives from a norm (providing a structure of n.v.s. on 𝕁et((M, a), (E, 0))).

2. Let M, M′ be metric spaces, a ∈ M, a′ ∈ M′; let also φ ∈ 𝕁et((M′, a′), (M, a)) and E a n.v.s. Then, the map φ̃: 𝕁et((M, a), (E, 0)) ⟶ 𝕁et((M′, a′), (E, 0)) : ψ ↦ ψ.φ is linear and continuous.

3. E and E′ being n.v.s., the canonical map : j : L(E, E′) ⟶ 𝕁et((E, 0), (E′, 0)) : l ↦ q(l) is a linear isometric embedding (L(E, E′) being equipped with the norm ∥l∥ = sup_{∥x∥≤1} ∥l(x)∥).

2.16. Corollary. If l : E ⟶ E′ is a continuous linear map, then q(l) is a good jet; more precisely, we have the equalities: d(q(l), O) = ρ(q(l)) = ∥l∥. Here, we can replace “linear” by “affine”.

3. Tangentiability

In this new context, the notion of tangentiability plays the part of the one of differentiability, of which it keeps lots of properties. This section gives some examples and counter-examples to understand and visualize this new notion.

3.1. Definition. Let f : M ⟶ M′ be a map between metric spaces and a ∈ M. We say that f is tangentiable at a (in short T ang_{a}) if there exists a map g : M ⟶ M′ which is LL_{a} such that g ≻≺_{a} f.

When f is T ang_{a}, we set Tf_{a} = {g : M ⟶ M′ | g ≻≺_{a} f ; g LL_{a}}; Tf_{a} is a jet (M, a) ⟶ (M′, f(a)), said tangent to f at a, and that we can call the tangential of f at a (not to be mixed up with tf_{a}, defined in a special context (see Section 5)).

3.2. Proposition. The inverses of the following implications are not true (see 3.7).

1. We have the implications: f LL_{a} ⟹ f T ang_{a} and f T ang_{a} ⟹ f LSL_{a} ⟹ continuous at a.

2. Let E, E′ be n.v.s., U an open subset of E, a ∈ U and f : U ⟶ E′ a map. We have the implication: f differentiable at a ⟹ f tangentiable at a.

3.3. Remarks.

1. f is LL_{a} iff f is T ang_{a} with f ∈ Tf_{a} (then Tf_{a} = q(f)); in particular, T(Id_{M})_{a} = q(Id_{M}) = Id_{(M,a)}. Moreover, for every jet φ : (M, a) ⟶ (M′, a′), we have φ = Tg_{a} for every g ∈ φ.

2. Actually, f is differentiable at a iff f is T ang_{a} where its tangential Tf_{a} at a possesses an affine map. For such a differential map, it is the unique affine map Af_{a} defined by Af_{a}(x) = f(a) + df_{a}(x − a)(the translate at a of its differential df_{a}); thus Tf_{a} = q(Af_{a}).

3.4. Proposition. (properties of the tangential)

1. Let M, M′, M′′ be metric spaces, f : M ⟶ M′, g : M′ ⟶ M′′ two maps, and a ∈ M, a′ = f(a). If f is T ang_{a} and g is T ang_{a′}, then g.f is T ang_{a} and we have T(g.f)_{a} = Tg_{a′}.Tf_{a}.

2. Let M, M_{0}, M_{1} be metric spaces, f_{0} : M ⟶ M_{0}, f_{1} : M ⟶ M_{1} be two maps, and a ∈ M. If f_{0} and f_{1} are tangentiable at a, then (f_{0}, f_{1}) : M ⟶ M_{0} × M_{1} is tangentiable at a and we have T(f_{0}, f_{1})_{a} = (Tf_{0a}, Tf_{1a}).

3. Let M_{0}, M_{1}, M′_{0}, M′_{1} be four metric spaces, f_{0} : M_{0} ⟶ M′_{0} and f_{1} : M_{1} ⟶ M′_{1} two maps, and a_{0} ∈ M_{0}, a_{1} ∈ M_{1}. If f_{0} is T ang_{a_{0}} and f_{1} T ang_{a_{1}}, then f_{0} × f_{1} : M_{0} × M_{1} ⟶ M′_{0} × M′_{1} is T ang_{(a_{0},a_{1})} and we have T(f_{0} × f_{1})_{(a_{0},a_{1})} = Tf_{0a_{0}} × Tf_{1a_{1}}.

3.5. Examples and counter-examples. All the maps considered below are functions ℝ ⟶ ℝ, where ℝ is equipped with its usual structure of normed vector space (these different functions give counter-examples to the inverses of the implications given in 3.2).

1. Consider f_{0}(x) = x^{1/3}; this function is continuous but not LSL_{0}.

2. Consider f_{1}(x) = x sin \frac{1}{x} if x ≠ 0 and f_{1}(0) = 0; this function is obviously LSL_{0}, however not T ang_{0}: indeed, if f_{1} was T ang_{0}, there would exist a fonction g : ℝ ⟶ ℝ and a neighborhood V of 0 such that g ≻≺_{0} f_{1} and g|_{V} is k-lipschitzian for a k > 0. Let us consider the two sequences of reals defined by x_{n} = 1/2nπ and y_{n} = 1/(4n + 1)\frac{π}{2}; they verify lim_{n} |\frac{f_{1}(x_{n})−g(x_{n})}{x_{n}}| = 0 = lim_{n} |\frac{f_{1}(y_{n})−g(y_{n})}{y_{n}}|, so that lim_{n} 2πng(x_{n}) = 0 and lim_{n}(4n + 1)\frac{π}{2}g(y_{n}) = 1. Now, since g|_{V} is k-lipschitzian, we have |\frac{g(x_{n})−g(y_{n})}{x_{n}−y_{n}}| ≤ k for n big enough, which is equivalent to |\frac{4n+1}{n}(2nπg(x_{n})) − 4((4n + 1)\frac{π}{2}g(y_{n}))| ≤ \frac{k}{n}. It remains to do n → +∞ which leads to a contradiction.

3. Consider f_{2}(x) = x^{2} sin \frac{1}{x^{2}} if x ≠ 0 and f_{2}(0) = 0; this function is T ang_{0} (since it is differentiable at 0); however not LL_{0} (because lim_{k→+∞}f′_{2}(\frac{1}{\sqrt{2kπ}}) = −∞).

4. Consider ϑ(x) = |x|; this function is T ang_{0} (it is 1-lipschitzian!), however not differentiable at 0.

Let us now consider two metric spaces M, M′, and two mapsf, g : M ⟶ M′ which are T ang_{a} where a ∈ M. As for general jets (see Section 2), we can speak of the distance between the two jets Tf_{a}, Tg_{a}. The question is: do we still have d(Tf_{a},Tg_{a}) = d(f, g)?, provided that we can define d(f, g) for maps f and g which are only T ang_{a}:

So, let M, M′ be metric spaces, a ∈ M, a′ ∈ M′ and f, g : M ⟶ M′ two maps which are T ang_{a} and which verify f(a) = g(a) = a′. Let us consider f_{1} ∈ Tf_{a} and g_{1} ∈ Tg_{a}; then f_{1} ≻≺_{a} f and g_{1} ≻≺_{a} g; so that there exists a neighborhood V of a on which we have d(f_{1}(x), f(x)) ≤ d(x, a) and d(g_{1}(x), g(x)) ≤ d(x, a). Furthermore, since f_{1} and g_{1} are LL_{a}, we know that there exists also a neighborhood W of a on which the map x ↦ \frac{d(f_{1}(x),g_{1}(x))}{d(x,a)}, if x ≠ a, is bounded (let us say by R). Now, if we take x ∈ V_{1} = V ∩ W,

x ≠ a, we obtain: \frac{d(f(x),g(x))}{d(x,a)} ≤ \frac{d(f(x),f_{1}(x))}{d(x,a)} + \frac{d(f_{1}(x),g_{1}(x))}{d(x,a)} + \frac{d(g_{1}(x),g(x))}{d(x,a)} ≤ 2 + R. So,

the map C(x) = \frac{d(f(x),g(x))}{d(x,a)}, if x ≠ a and C(a) = 0, is still bounded on V_{1}. Thus, for r > 0, we can again set d^{r}(f, g) = sup{C(x)|x ∈ V_{1} ∩ B′(a, r)} and finally again d(f, g) = lim_{r→0} d^{r}(f, g) = inf_{r>0} d^{r}(f, g).

3.6. Proposition. If f, g : M ⟶ M′ are T ang_{a} where a ∈ M, we have d(Tf_{a}, Tg_{a}) = d(f, g), this d being defined just above.

4. Transmetric spaces

In order to define a tangential map tf : M ⟶ “𝕁et(M, M′)” for a tangentiable map f : M ⟶ M′ (see Section 5), we need to define first such a set “𝕁et(M, M′)” equipped with an adequate distance. We have succeeded in it, assuming that M and M′ are transmetric spaces and introducing new jets called free metric jets (in opposition to the metric jets we have used up to now) whose set will be denoted 𝕁et_{free}(M, M′); this set being a good candidate for our unknown “𝕁et(M, M′)”.

4.1. Definition. A transmetric space is a metric space M, supposed to be non empty, equipped with a functor γ : Gr(M) ⟶ 𝕁et (where the category 𝕁et has been defined at the beginning of Section 2, and Gr(M) is the groupoid associated to the undiscrete equivalence relation on M; thus |Gr(M)| = M and Hom(a, b) = {(a, b)} for all a, b ∈ M) verifying:

- for every a ∈ M, γ(a) = (M, a),

- for every morphism (a, b) : a ⟶ b in Gr(M), the jet γ(a, b) : (M, a) ⟶ (M, b) is 1-bounded (i.e. verifies ρ(γ(a, b)) ≤ 1: see Section 2); thus invertible in 𝕁et.

Before giving some examples, let us give the following special case:

4.2. Definition. A left isometric group is a metric space G equipped with a group structure verifying the following condition: ∀g, g_{0}, g_{1} ∈ G (d(g.g_{0}, g.g_{1}) = d(g_{0}, g_{1})).

4.3. Remarks.

1. In an equivalent manner, in 4.2 we could assume only that d(g.g_{0}, g.g_{1}) ≤ d(g_{0}, g_{1}) for all g, g_{0}, g_{1} ∈ G.

2. Let G be a left isometric group; then, for all g ∈ G, the map G ⟶ G : g′ ↦ g.g′ is isometric.

4.4. Proposition. Every left isometric group G can be equipped with a canonical structure of transmetric space.

\underline{Proof} : Here, γ is the composite Gr(G)⟶^{θ}𝕃L ⟶^{q}𝕁et, where θ is the functor defined by θ(g) = (G, g) and θ(g_{0}, g_{1}) is the morphism (G, g_{0}) ⟶ (G, g_{1}) in 𝕃L which assignes g_{1}.g_{0}^{−1}.g to g. □

4.5. Examples.

1. Here are examples of transmetric spaces which are even left isometric groups:

(a) Every n.v.s. is a left isometric (additive) group: here, γ(a, b) = q(θ(a, b)), where θ(a, b) is the translation x ↦ b − a + x.

(b) The multiplicative group S^{1} = {z ∈ ℂ| |z| = 1} is also a left isometric group.

(c) E being an euclidian space, the orthogonal group O(E), equipped with its operator norm, is a left isometric group.

(d) The additive subgroups of ℝ (as, for example ℚ) are left isometric groups whuch are not n.v.s.

2. Every non empty discrete space has a unique structure of transmetric space; conversely, if a transmetric space possesses an isolated point, it is a discrete space.

3. The 4.6 below will provide a lot of transmetric spaces which are not left isometric groups.

4.6. Proposition.

1. Let M_{0} and M_{1} be transmetric spaces; then M_{0} × M_{1} has a canonical stucture of transmetric space.

2. Let M be a transmetric space and U ≠ ∅, an open subset of M; then, U has a canonical structure of transmetric space.

\underline{Proof} : 1. The functor γ : Gr(M_{0} × M_{1}) ⟶ 𝕁et is defined by γ((a_{0}, a_{1}), (b_{0}, b_{1})) = γ_{0}(a_{0}, b_{0}) × γ_{1}(a_{1}, b_{1}), where γ_{0} and γ_{1} give the transmetric structures on M_{0} and M_{1}.

2. The functor γ̆ : Gr(U) ⟶ 𝕁et is defined by γ̆(a) = (U, a) and γ̆(a, b) = j_{b}^{−1}.γ(a, b).j_{a} : (U, a) ⟶ (U, b). □

4.7. Definition. Let M, M′ be transmetric spaces; a map f : M ⟶ M′ is called a morphism of transmetric spaces if it is a tangentiable map (at every point of M) such that, for every a, b ∈ M, the following diagram commutes in the category 𝕁et:

\xymatrix{(M, a) \ar[d]_{γ(a,b)} \ar[r]^{Tf_{a}} & (M′, f(a)) \ar[d]^{γ(f(a),f(b))} \\ (M, b) \ar[r]_{Tf_{b}} & (M′, f(b))}

We will denote 𝕋rans the category whose objects are the transmetric spaces and whose morphisms are the morphisms of transmetric spaces.

4.8. Examples. (of such morphisms)

1. Every continuous affine map f : E ⟶ E′ (between n.v.s.).

2. Id_{M} : M ⟶ M and the canonical injection j : U ↪ M, if U ≠ ∅, is an open subset of a transmetric space M.

3. Every constant map (on c ∈ M′ ) ĉ: M ⟶ M′ between transmetric spaces; it is the case for the unique map !_{M} : M ⟶ 𝕀 (where M is a transmetric space and 𝕀 = {0}).

4. The canonical projections p_{i} : M_{0} × M_{1} ⟶ M_{i}, where M_{0} and M_{1} are transmetric spaces.

4.9. Proposition. The category 𝕋rans is a cartesian category, and the forgetful functor 𝕋rans ⟶ 𝔼ns is a morphism of cartesian categories.

We are now ready to construct the set 𝕁et_{free}(M, M′) of free metric jets, when M and M′ are transmetric spaces. First, we consider the set J(M, M′) = ∐_{(a,a′)∈M×M′} 𝕁et((M, a), (M′, a′)) on which we define the following equivalence relation: (φ, a, a′) ∼ (ψ, b, b′) if the following diagram commutes in the category 𝕁et:

\xymatrix{(M, a) \ar[d]_{γ(a,b)} \ar[r]^{φ} & (M′, a′) \ar[d]^{γ(a′,b′)} \\ (M, b) \ar[r]_{ψ} & (M′, b′)}

Then, we set 𝕁et_{free}(M, M′) = J(M, M′)/ ∼.

4.10. Definition. The elements of 𝕁et_{free}(M, M′) are called free metric jets (in short, free jets, here from M to M′).

If q : J(M, M′) ⟶ 𝕁et_{free}(M, M′) is the canonical surjection, we set [φ, a, a′] = q(φ, a, a′) when (φ, a, a′) ∈ J(M, M′).

We are going to canonically equip 𝕁et_{free}(M, M′) with a structure of metric space. First, for (φ, a, a′), (ψ, b, b′) ∈ J(M, M′), we denote d((φ, a, a′), (ψ, b, b′)) the distance between the jets γ(a′, b′).φ and ψ.γ(a, b) (see Section 2). Using the properties of the lipschitzian ratio and the theorem of Section 2, we prove:

4.11. Proposition.

1. For each (φ, a, a′), (ψ, b, b′), (ξ, c, c′) ∈ J(M, M′), we have the following properties:

(a) d((φ, a, a′), (ψ, b, b′)) = d((ψ, b, b′), (φ, a, a′)),

(b) d((φ, a, a′), (ξ, c, c′)) ≤ d((φ, a, a′), (ψ, b, b′)) + d((ψ, b, b′), (ξ, c, c′)),

(c) d((φ, a, a′), (ψ, b, b′)) = 0 ⟺ (φ, a, a′) ∼ (ψ, b, b′).

2. The map d : (J(M, M′))^{2} ⟶ ℝ_{+} factors through the quotient, giving a “true” distance on 𝕁et_{free}(M, M′), defined by d([φ, a, a′], [ψ, b, b′]) = d((φ, a, a′), (ψ, b, b′)) for all (φ, a, a′), (ψ, b, b′) ∈ J(M, M′).

4.12. Remark. Let M, M′ be transmetric spaces; then 𝕁et_{free}(M, M′) possesses a particular element [O_{aa′}, a, a′] denoted O: for [O_{aa′}, a, a′] does not depend on the choice of (a, a′) ∈ M × M′; this free jet O will be called the free zero of 𝕁et_{free}(M, M′). In the same way, 𝕁et_{free}(M, M) possesses also a free identity, denoted I_{M}, which is equal to [Id_{(M,x)}, x, x].

4.13. Proposition. Let M, M′ be transmetric spaces, a, b ∈ M and a′, b′ ∈ M′.

1. If c(φ) = (φ, a, a′), then the composite

can : 𝕁et((M, a), (M′, a′))⟶^{c}J(M, M′)⟶^{q}𝕁et_{free}(M, M′)

is an isometry.

2. Let us set Ω(φ) = γ(a′, b′).φ.γ(b, a); this defines a map

Ω : 𝕁et((M, a), (M′, a′)) ⟶ 𝕁et((M, b), (M′, b′))

which is an isometry, and the following diagram commutes:

\xymatrix{𝕁et((M, a), (M′, a′)) \ar[rr]^{Ω} \ar@{-->}[dr]_{can} &  & 𝕁et((M, b), (M′, b′)) \ar@{-->}[dl]^{can} \\  & 𝕁et_{free}(M, M′)}

Let now M_{0}, M_{1}, M_{2} be three transmetric spaces; we consider the map

comp : J(M_{0}, M_{1}) × J(M_{1}, M_{2}) ⟶ J(M_{0}, M_{2})

defined by (φ_{1}, b_{1}, b_{2}).(φ_{0}, a_{0}, a_{1}) = (φ_{1}.γ(a_{1}, b_{1}).φ_{0}, a_{0}, b_{2}) (as in the diagram below):

\xymatrix{(M_{0}, a_{0}) \ar[r]^{φ_{0}} & (M_{1}, a_{1}) \ar[r]^{γ(a_{1},b_{1})} & (M_{1}, b_{1}) \ar[r]^{φ_{1}} & (M_{2}, b_{2})}

4.14. Proposition.

1. This map comp factors through the quotient:

\xymatrix{J(M_{0}, M_{1}) × J(M_{1}, M_{2}) \ar[d]_{q×q} \ar[r]^{comp} & J(M_{0}, M_{2}) \ar[d]^{q} \\ 𝕁et_{free}(M_{0}, M_{1}) × 𝕁et_{free}(M_{1}, M_{2}) \ar[r]_{comp} & 𝕁et_{free}(M_{0}, M_{2})}

(and we will write [φ_{1}, b_{1}, b_{2}].[φ_{0}, a_{0}, a_{1}] = [φ_{1}.γ(a_{1}, b_{1}).φ_{0}, a_{0}, b_{2}], which gives simply

[φ_{1}, b_{1}, b_{2}].[φ_{0}, a_{0}, a_{1}] =^{⋆} [φ_{1}.φ_{0}, a_{0}, b_{2}] when a_{1} = b_{1}).

2. The composition comp : 𝕁et_{free}(M_{0}, M_{1}) × 𝕁et_{free}(M_{1}, M_{2}) ⟶ 𝕁et_{free}(M_{0}, M_{2}), defined just above, is LSL.

So, we construct a new category, enriched in 𝕄et, denoted 𝕁et_{free}, called the category of free jets,whose:

- objects are the transmetric spaces M,

- “Hom” are the metric spaces 𝕁et_{free}(M, M′),

- identity 𝕀 ⟶ 𝕁et_{free}(M, M), is the map giving the free identity I_{M} = [Id_{(M,a)}, a, a],

- composition 𝕁et_{free}(M, M′)×𝕁et_{free}(M′, M′′) ⟶ 𝕁et_{free}(M, M′′) is the previous comp.

4.15. Definition. We will denote 𝕁et′ the full subcategory of the category 𝕁et whose objects are the pointed transmetric spaces. Just as 𝕁et (see Section 2), it is a category enriched in 𝕄et.

4.16. Proposition.

1. Actually, we have a forgetful enriched functor U : 𝕁et′ ⟶ 𝕁et.

2. We have a functor can′ : 𝕁et′ ⟶ 𝕁et_{free}, defined by can′(M, a) = M and, for φ : (M, a) ⟶ (M′, a′), by can′(φ) = can(φ) = [φ, a, a′]; then, this functor is enriched in 𝕄et.

We now come back to cartesian considerations.

4.17. Remark. By definition, f : M ⟶ M′ is a morphism of transmetric spaces iff we have (Tf_{x}, x, f(x)) ∼ (Tf_{y}, y, f(y)) for all x, y ∈ M; so that the free jet [Tf_{x}, x, f(x)] is independent on the choice of x ∈ M; we denote κ(f) this element of 𝕁et_{free}(M, M′). In particular, if c ∈ M′, we have seen that the constant map ĉ is a morphism of transmetric spaces and that κ(ĉ) = O.

A METRIC TANGENTIAL CALCULUS  215

4.18. Proposition.

1. The map 𝕋rans(M, M′) ⟶ 𝕁et_{free}(M, M′) : f ↦ κ(f) extends to a functor κ : 𝕋rans ⟶ 𝕁et_{free} which is constant on the objects.

2. This functor κ : 𝕋rans ⟶ 𝕁et_{free} creates a cartesian structure on the category 𝕁et_{free} (κ being constant on the objects, it means that 𝕁et_{free} is cartesian and κ a strict morphism of cartesian categories).

3. The functor can′ : 𝕁et′ ⟶ 𝕁et_{free} is a morphism of cartesian categories.

4. M, M_{0}, M_{1} being transmetric spaces, the canonical map

can : 𝕁et_{free}(M, M_{0} × M_{1}) ⟶ 𝕁et_{free}(M, M_{0}) × 𝕁et_{free}(M, M_{1})

is an isometry (like its analogue for 𝕁et). 4.19. Proposition. Let M, M′ be two transmetric spaces, and U, U′ two non empty open subsets of M and M′ respectively (with j : U ↪ M and j′ : U′ ↪ M′ the canonical injections).

1. κ(j) : U ⟶ M is an isomorphism in 𝕁et_{free}.

2. The canonical map Γ : 𝕁et_{free}(U, U′) ⟶ 𝕁et_{free}(M, M′) : Φ ↦ κ(j′).Φ.κ(j)^{−1} is an

isometry. 4.20. Proposition.

1. Let M, M′ be two transmetric spaces. Then the map J(M, M′) ⟶ ℝ_{+} : (φ, a, a′) ↦ ρ(φ) factors through the quotient; we still denote ρ : 𝕁et_{free}(M, M′) ⟶ ℝ_{+} this factorization. (see Section 2 for the definition of the lipschitzian ratio of a jet).

2. Let M_{0}, M_{1}, M_{2} be three transmetric spaces; then, if Φ_{0} : M_{0} ⟶ M_{1} and Φ_{1} : M_{1} ⟶ M_{2} are free jets, we have ρ(Φ_{1}.Φ_{0}) ≤ ρ(Φ_{1})ρ(Φ_{0}).

Again, we conclude this paragraph with vectorial considerations. 4.21. Proposition.

1. Let M be a transmetric space and E a n.v.s. Then, the metric space 𝕁et_{free}(M, E) has also a canonical structure of n.v.s. (its distance defined in 4.11 derives from a norm). Besides, for every a ∈ M, the map can : 𝕁et((M, a), (E, 0)) ⟶ 𝕁et_{free}(M, E) (see 4.13) is a linear isometry.

2. Let M be a transmetric space and E a n.v.s. Let also Φ ∈ 𝕁et_{free}(M′, M). Then, the map Φ̃ : 𝕁et_{free}(M, E) ⟶ 𝕁et_{free}(M′, E) : Ψ ↦ Ψ.Φ is linear and continuous.

5. Tangential

Untill now, we have only spoken of the tangential Tf_{a}, at a, of a map f : M ⟶ M′, tangentiable at a (see Section 3). ¿From now on, we will speak of the tangential tf : M ⟶ 𝕁et_{free}(M, M′) when f : M ⟶ M′ is a tangentiable map between transmetric spaces (as we speak of the differential df : U ⟶ L(E, E′) for a differentiable map f : U ⟶ E′ when U is an open subset of E, a n.v.s., just as E′).

First, consider M, M′ two transmetric spaces and f : M ⟶ M′ a tangentiable map at the point x ∈ M; then, we set tf_{x} = [Tf_{x}, x, f(x)].

Now, if f : M ⟶ M′ is tangentiable (at every point in M), we can define the map tf : M ⟶ 𝕁et_{free}(M, M′) : x ↦tf_{x} (in fact, this map tf is the following composite

M ⟶^{Tf} J(M, M′) ⟶^{q} 𝕁et_{free}(M, M′), where J(M, M′), 𝕁et_{free}(M, M′) and q have been defined in Section 4, and where Tf(x) = (Tf_{x}, x, f(x)).

5.1. Definition. The map tf defined just above (for a tangentiable map f between transmetric spaces) will be called the tangential of f.

5.2. Proposition. Let f : M ⟶ M′ be a tangentiable map between transmetric spaces. Then, the map tf : M ⟶ 𝕁et_{free}(M, M′) is constant iff f is a morphism of transmetric spaces; in this case we have tf_{x} = κ(f) for every x ∈ M.

Let us now begin with the particular vectorial context: we assume here that E, E′ are n.v.s. and U a non empty open subset of E; consider then the following composite:

\xymatrix{J : L(E, E′) \ar[r]^{j} & 𝕁et((E, 0), (E′, 0)) \ar[r]^{can} & 𝕁et_{free}(E, E′) \ar[r]^{Γ^{−1}} & 𝕁et_{free}(U, E′)}

See Section 2 for j and Section 4 for can and Γ. We set Im(J) = J(L(E, E′)), the image of J in 𝕁et_{free}(U, E′).

5.3. Remark. The map J : L(E, E′) ⟶ 𝕁et_{free}(U, E′) defined just above is a linear isometric embedding.

5.4. Proposition.

1. Let f : U ⟶ E′ be a tangentiable map and a ∈ U; then, f is differentiable at a iff tf_{a} ∈ Im(J); in this case, we have tf_{a} = J(df_{a}).

2. Let f : U ⟶ E′ a differentiable map; then f is tangentiable and the following diagram commutes:

\xymatrix{ & U \ar[dl]_{df} \ar[dr]^{tf} \\ L(E, E′) \ar[rr]_{J} &  & 𝕁et_{free}(U, E′)}

The proof of 5.4.1 uses the result of the following lemma:

5.5. Lemma. Let [φ, a, b] ∈ 𝕁et_{free}(U, E′). Then, [φ, a, b] ∈ Im(J) iff there exists l ∈ L(E, E′) such that Al|_{U} ∈ φ, where Al(x) = b + l(x − a); and we have J(l) = [φ, a, b].

We now come back to the general transmetric context.

5.6. Proposition. Let M, M′, M′′ be three transmetric spaces; f : M ⟶ M′, g : M′ ⟶ M′′ two maps. We assume that f and g are tangentiable and that their tangentials tf : M ⟶ 𝕁et_{free}(M, M′) and tg : M′ ⟶ 𝕁et_{free}(M′, M′′) are continuous, respectively at a ∈ M and a′ = f(a) ∈ M′. Then the map t(g.f) : M ⟶ 𝕁et_{free}(M, M′′) is well defined and continuous at a, and we have t(g.f)_{a} = tg_{a′}.tf_{a}.

\underline{Proof} : The continuity of the map t(g.f) comes from the following composite:

\xymatrix{M \ar[r]^{(Id,f)} & M × M′ \ar[r]^{tf×tg} & 𝕁et_{free}(M, M′) × 𝕁et_{free}(M′, M′′) \ar[r]^{comp} & 𝕁et_{free}(M, M′′)}
□

5.7. Proposition. Let M, M_{0}, M_{1} be transmetric spaces, with a ∈ M and f_{0} : M ⟶ M_{0}, f_{1} : M ⟶ M_{1} two maps. We assume that, for each i ∈ {0, 1}, f_{i} : M ⟶ M_{i} is tangentiable and that its tangential tf_{i} : M ⟶ 𝕁et_{free}(M, M_{i}) is continuous at a. Then, the map t(f_{0}, f_{1}) : M ⟶ 𝕁et_{free}(M, M_{0} × M_{1}) is well defined and continuous at a, and we have t(f_{0}, f_{1})_{a} = (tf_{0a}, tf_{1a}).

\underline{Proof} : The continuity of the map t(f_{0}, f_{1}) comes from the following composite:

\xymatrix{M \ar[r]^{(tf_{0},tf_{1})} & 𝕁et_{free}(M, M_{0}) × 𝕁et_{free}(M, M_{1}) \ar[r]^{can^{−1}} & 𝕁et_{free}(M, M_{0} × M_{1})}

□

5.8. Proposition. Let E be a n.v.s., U an open subset of E, a, b ∈ U such that [a, b] ⊂ U and F a finite subset of ]a, b[; let also M be a transmetric space and f : U ⟶ M a continuous map. Now, if we assume that, for all x ∈]a, b[−F, the map f is T ang_{x} and that d(tf_{x}, O) ≤ k (where O is the free zero of 4.12 and k a fixed positive real number), then we have d(f(b), f(a)) ≤ k||b − a||.

\underline{Proof} : Comes from 1.7 and the following lemma:

5.9. Lemma. Let f : M ⟶ M′ be a map, a ∈ M and b = f(a). We assume that f is T ang_{a} and that d(Tf_{a}, O_{ab}) < k. Then, f is k-LSL_{a} (thanks to 3.6).

We now give a generalization of 1.6.4.(b):

5.10. Theorem. Let E and M be respectively a n.v.s. and a transmetric space, U a non empty open subset of E and f : U ⟶ M a tangentiable map such that tf : U ⟶ 𝕁et_{free}(U, M) is continuous at a ∈ U. Then, f is LL_{a} and we have d(tf_{a}, O) = ρ(tf_{a}); i.e tf_{a} is a good free jet (the analogue, for the free jets, of what we have previously defined for the jets in Section 2).

\underline{Proof} : First, thanks to 2.6.2 and by definition of the lipschitzian ratio ρ for jets or free metric jets (see 2.5 and 4.25), we have d(tf_{a}, O) = d(Tf_{a}, O_{af(a)}) ≤ ρ(Tf_{a}) = ρ(tf_{a}).

Now, let f̄: U ⟶ ℝ be the following composite:

\xymatrix{U \ar[r]^{tf} & 𝕁et_{free}(U, M) \ar[r]^{d(−,O)} & ℝ}

The map f̄ is continuous at a, by composition. Let us fix ε > 0; then, there exists r > 0 such that B(a, r) ⊂ U and f̄(x) < f̄(a) + ε for all x ∈ B(a, r). Let us set R = f̄(a) + ε; then, for all x ∈ B(a, r), we have d(tf_{x}, O) = f̄(x) < R, so that (thanks to 5.8), the restriction f|_{B(a,r)} is R-lipschitzian (since B(a, r) is convex). Thus f is LL_{a}, so that Tf_{a} = q(f). Furthermore, we have ρ(tf_{a}) = ρ(Tf_{a}) = ρ(q(f)) ≤ R = f̄(a) + ε; we have thus obtained ρ(tf_{a}) ≤ d(tf_{a}, O) + ε for all ε > 0, so that ρ(tf_{a}) ≤ d(tf_{a}, O). □

5.11. Proposition. Let E be a n.v.s. and U a non empty open subset of E, and M a transmetric space.

1. If U is convex, then every morphism of transmetric spaces f : U ⟶ M is R-lipschitzian (where R = d(κ(f), O); see Section 4 for κ(f)). Hence, d(κ(f), O) = ρ(κ(f)); i.e κ(f) is a good free jet (this result is true for every continuous affine map).

2. If U is not convex, f is R-LL on U, and κ(f) is still a good free jet.

\underline{Proof} : 1. We use 5.8, since, for all x ∈ U, we have d(tf_{x}, O) = d(κ(f), O) = R; thus f is R-lipschitzian on U (which is convex), so that ρ(κ(f)) ≤ R. For the inverse inequality, we use 2.6.2.

2. Because every open subset of a n.v.s. is locally convex. □

5.12. Definition. Let M, M′ be transmetric spaces and f : M ⟶ M′ a map. We say that f is continuously tangentiable if f is tangentiable and if its tangential tf : M ⟶ 𝕁et_{free}(M, M′) is continuous.

5.13. Proposition. The continuously tangentiable maps are stable under composition and under pairs.

5.14. Examples. (of continuously tangentiable maps)

1. Every morphism of transmetric spaces.

2. Every map which is of class C^{1}.

5.15. Theorem. Let E, E′ be n.v.s. where E′ is complete, U a non empty open subset of E and f : U ⟶ E′ a continuously tangentiable map which is also differentiable at every point of a dense subset D of U. Then, f is of class C^{1}.

\underline{Proof} : Referring to 5.4.1, we know that, a priori, the set of the points of U at which

f is differentiable is nothing but tf^{−1} (Im(J)); thus, we must first show that this set is equal to U. Since, by hypothesis, we have the inclusions D ⊂ tf^{−1} (Im(J)) ⊂ U with D

dense in U, we have just to prove that tf^{−1} (Im(J)) is closed in U. Actually, since Im(J) is complete (just as the n.v.s. L(E, E′), for J : L(E, E′) ⟶ 𝕁et_{free}(U, E′) is an isometric

embedding), it is closed in 𝕁et_{free}(U, E′), so that tf^{−1} (Im(J)) is also closed in U (since tf : U ⟶ 𝕁et_{free}(U, E′) is continuous); f is thus differentiable on U. Finally, the fact that the map df : U ⟶ L(E, E′) is continuous comes from the fact it factors through tf (see the commutative triangle in 5.4.2. □

5.16. Example. The function ϑ : ℝ ⟶ ℝ : x ↦ |x| is tangentiable (for it is lipschitzian); however, it cannot be continuously tangentiable on ℝ, since it is differentiable on ℝ^{∗} and not of class C^{1} on ℝ.

We now give a generalization of the fact that the composite of two maps of class C^{2} is also of class C^{2}.

5.17. Theorem. Let M, M′ be transmetric spaces, E a n.v.s. and U a non empty open subset of E. Let also f : M ⟶ U and g : U ⟶ M′ be two continuously tangentiable maps. We assume that tf : M ⟶ 𝕁et_{free}(M, U) is T ang_{a} where a ∈ M and that tg : U ⟶ 𝕁et_{free}(U, M′) is LL_{a′} where a′ = f(a) ∈ U. Then, t(g.f) : M ⟶ 𝕁et_{free}(M, M′) is T ang_{a}.

\underline{Proof} : Since g : U ⟶ M′ is continuously tangentiable, we know (by 5.10) that d(tg_{y}, O) = ρ(tg_{y}) for all y ∈ U. We now need the following lemma:

5.18. Lemma.

1. Let (M_{1}, a_{1}), (M_{2}, a_{2}) ∈ |𝕁et| and Jeg((M_{1}, a_{1}), (M_{2}, a_{2})) be the set of good jets, i.e {φ ∈ 𝕁et ((M_{1}, a_{1}), (M_{2}, a_{2})) |d(φ, O_{a_{1}a_{2}}) = ρ(φ)}. Then the following restriction is LL: 𝕁et((M_{0}, a_{0}),(M_{1}, a_{1}))! ×Jeg((M_{1}, a_{1}),(M_{2}, a_{2})) ⟶^{comp} 𝕁et((M_{0}, a_{0}),(M_{2}, a_{2})).

2. Let M_{0}, M_{1}, M_{2} be transmetric spaces;  if Jeg_{free}(M_{1}, M_{2})  =

= {Φ ∈ 𝕁et_{free}(M_{1}, M_{2}) | d(Φ, O) = ρ(Φ)}. Then the following restriction is LL:

𝕁et_{free}(M_{0}, M_{1}) × Jeg_{free}(M_{1}, M_{2}) ⟶^{comp} 𝕁et_{free}(M_{0}, M_{2}).

Thus, coming back to the proof of 5.17, we deduce that tg_{y} ∈ Jeg_{free}(U, M′). We can then consider the restriction t̆g : U ⟶ Jeg_{free}(U, M′) of tg. Besides, knowing that g.f is continuously tangentiable with t(g.f)_{x} =tg_{f(x)}.tf_{x} for all x ∈ M, we can write its tangential t(g.f) as the composite:

\xymatrix{M \ar[r]^{(Id,f)} & M × U \ar[r]^{tf×t̆g} & 𝕁et_{free}(M, U) × Jeg_{free}(U, M′) \ar[r]^{comp} & 𝕁et_{free}(M, M′)}

So that we only have to justify the tangentiability, at the right point, of every forementionned map: by Section 3, (Id, f) is T ang_{a} and tf × t̆g is T ang_{(a,a′)}; and comp is LL_{(tf_{a},tg_{a′})} (by lemma 5.18), thus T ang_{(tf_{a},tg_{a′})}. Their composite t(g.f) is thus T ang_{a}. □

Here are some open problems that we submit to the sagacity of the reader (after having gone through this paper)

1. We have seen, in the examples given in 4.8, that every continuous affine map between n.v.s. is a morphism of transmetric spaces. The question is: are there any other morphisms of transmetric spaces in such a vectorial context?

2. Still in the case of the n.v.s., does there exist continuously tangentiable maps which are not of class C^{1} (see 5.14 and 5.15)?

References

[1] F. Borceux, Handbook of Categorical Algebra, Cambridge University Press (1994).

[2] E. Burroni and J. Penon, Arrêt sur la tangentialité, talk at SIC in Calais (2006).

[3] E. Burroni and J. Penon, Un calcul différentiel métrique, talk at CT08 in Calais (2008).

[4] E. Burroni and J. Penon, Elements for a metric tangential calculus, http://fr.arxiv.org/abs/0912.1012 (2009).

[5] C. Ehresmann, Les prolongements d’une variété différentiable, I – V, CRAS Paris 233 (1951) 598-600, 777-779, 1081-1083, CRAS Paris 234 (1952) 1028-1030, 1424-1425.

Institut de Mathématiques de Jussieu; Université Paris Diderot, Paris 7; 5, rue Thomas Mann; 75205 Paris Cedex 13; France.

Email: eburroni@math.jussieu.fr, penon@math.jussieu.fr

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