Angles between curves in metric measure spaces
Bang-Xian Han, Andrea Mondino

TL;DR
This paper introduces a new way to define angles between curves in metric measure spaces, linking it with optimal transportation and curvature conditions, and proves the cosine formula in certain curvature-dimension spaces.
Contribution
It proposes a novel notion of angle in metric spaces that aligns with classical concepts and applies to spaces with Ricci curvature bounds.
Findings
Validates the cosine formula in $RCD^{*}(K,N)$ spaces
Ensures compatibility with classical angles in Riemannian and Alexandrov spaces
Connects angle notions with optimal transportation theory
Abstract
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.
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Angles between curves in metric measure spaces
Bang-Xian Han
and
Andrea Mondino
Abstract.
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.
University of Bonn, [email protected]
University of Warwick, [email protected]
Keywords: angle, metric measure space, Wasserstein space.
Contents
1. Introduction
The ‘angle’ between two curves is a basic concept of mathematics, which aims to quantify the infinitesimal distance between two crossing curves at a crossing point. Such a notion is classical in Euclidean and in Riemannian geometries where a global (respectively infinitesimal) scalar product is given: the cosine of the angle between two crossing curves is by definition the scalar product of the velocity vectors. If the space is not given an infinitesimal scalar product, it is a challenging problem to define angles in a sensible way. In this paper, we will study this problem in a metric (measure) sense. More precisely, consider a metric space , a point , and two geodesics such that . Our task is to propose a meaningful definition of the angle between the curves at the point , denoted by , and to establish some interesting properties.
We recall some examples first. Assume that and are geodesics, and the space is an Alexandrov space, with upper or lower curvature bounds. From the monotonicity implied by the Alexandrov condition, it is known (see for instance [7]) that the angle is well defined by the cosine formula:
[TABLE]
In order to define the angle for geodesics in a more general framework, a crucial observation is that a geodesic can be seen as gradient flow of the distance function, i.e. a geodesic ‘represents’ the gradient of on each point . Inspired by the seminal work of De Giorgi on gradient flows [15], given an arbitrary metric space with a geodesic and a Lipschitz function , we say that represents at time [math], or represents the gradient of the function at the point if the following inequality holds
[TABLE]
where is the (constant, metric) speed of the geodesic . Notice that the opposite inequality
[TABLE]
is always true by Leibniz rule and Cauchy-Schwartz inequality. Hence represents at time [math] if and only if the equality holds. It is easily seen that the geodesic always represents the gradient of at the point (see for instance Lemma 3.5). We then say that the angle between two geodesics with exists if the limit exists. In this case we set
[TABLE]
Notice that in case is the metric space associated to a smooth Riemannian manifold , the definition (1.1) reduces to the familiar notion of angle
[TABLE]
Besides the case of Alexandrov spaces, a class of spaces where the angle is particularly well behaved is the one of Lipschitz-infinitesimally Hillbertian spaces. By definition, a metric measure space is Lipschitz-infinitesimally Hillbertian if for any pair of Lipschitz functions both the limits for of and exist and are equal for -a.e. , where is the local Lipschitz constant of (for the standard definition see (2.1)). A remarkable example of Lipschitz-infinitesimally Hillbertian spaces is given by the -spaces, a class of metric measure spaces satisfying Ricci curvature lower bound by and dimension upper bound by in a synthetic sense such that the Laplacian is linear, and which include as notable subclasses the Alexandrov spaces with curvature bounded below and the Ricci limit spaces (i.e. pointed measured Gromov-Hausdorff limits of sequences of Riemannian manifolds with uniform lower Ricci curvature bounds).
In the class of Lipschitz-infinitesimally Hillbertian spaces, the second author [26] introduced a notion of ‘angle between three points’; more precisely for every fixed pair of points , for -a.e. the angle given by the formula
[TABLE]
is well defined, unique, and symmetric in and . Here is the distance function from . A first result of the present paper is to relate the angle between three points with the angle between two geodesics: in Theorem 3.9 we prove that if the angle exists in the sense of [26] then also the angle between the geodesics joining to and to exists and coincides with the angle between the three points, i.e. . In particular it follows that in a Lipschitz-infinitesimally Hilbertian geodesic space the angle between two geodesics in well defined in an a.e. sense.
An important class of metric spaces are the spaces of probability measures over metric spaces endowed with the quadratic transportation distance: given a metric space denote by the corresponding Wasserstein space. By using ideas similar to the ones above, together with Otto Calculus (see [28]) and the calculus tools developed by Ambrosio-Gigli-Savaré [2] and Gigli [17], in Subsection 3.3 we study in detail the angle between two geodesics in . In particular if the underlying space is an space, we get the angle between three points as the limit of the angle between the geodesics in obtained by joining geodesically diffused approximations of Dirac masses centered at , and (see Proposition 3.15 for the precise statement; see also Proposition 3.17 for a more detailed link with the optimal transport picture).
Besides the case of Alexandrov spaces, another class of spaces where the notion of angle is quite well understood is given by Ricci limit spaces. Indeed it was proved by Honda [23] that if is a Ricci-limit space, then for -a.e. the angle between two geodesics is well defined and it satisfies the following one-variable cosine formula:
[TABLE]
One of the main goals of the present paper is to extend the validity of the formula (1.3) to metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, the so-called -spaces (for the definition and basic properties of such spaces see Section 2 and references therein). This is the content of the next theorem (corresponding to Theorem 4.4 in the body of the manuscript), which is one of the main results of the paper.
Theorem 1.1** (Cosine formula for angles in spaces).**
Let be an space and fix . Then for -a.e. there exist unique geodesics from to and from to denoted by and
[TABLE]
The proof of Theorem 1.1 in independent and different from the one of Honda [23] for Ricci limit spaces: indeed Honda argues by getting estimates on the smooth approximating manifolds and then passes to the limit, while our proof for spaces goes by arguing directly on the non smooth space . More precisely, we perform a blow up argument centered at and use that for -a.e. the tangent cone is unique and euclidean [21, 27]. From the technical point of view we also make use of the fine convergence results for Sobolev functions proved in [20, 4], and we prove estimates on harmonic approximations of distance functions (see in particular Proposition 4.3). Harmonic approximations of distance functions are well known for smooth Riemannian manifolds with lower Ricci curvature bounds, and are indeed one of the key technical tools in the Cheeger-Colding theory of Ricci limit spaces [12, 13, 14]; on the other hand for non-smooth -spaces it seems they have not yet appeared in the literature, and we expect them to be a useful technical tool in the future development of the field.
As a consequence of Theorem 1.1, we get that our definition of angle between two geodesics agrees (at least in a.e. sense) with the Alexandrov’s definition in case is an Alexandrov space, and with the Honda’s definition [23] in case is a Ricci limit space.
Acknowledgement: The first author would like to thank Nicola Gigli, Shouhei Honda and Karl Theodor Sturm for discussions on the topic.
2. Preliminaries
2.1. Metric measure spaces
Let be a complete metric space. A continuous map will be called curve. The space of curves defined on with values in is denoted by . The space equipped with the uniform distance is a complete metric space.
We define the length of by
[TABLE]
where is a partition of , and the is taken over all finite partitions. The space is said to be a length space if for any we have
[TABLE]
where the infimum is taken over all connecting and . A geodesic from to is a curve such that:
[TABLE]
The space of all geodesics on will be denoted by . It is a closed subset of .
Given and a curve , we say that belongs to if
[TABLE]
for some . In particular, the case corresponds to absolutely continuous curves, whose class is denoted by . It is known that for , there exists an a.e. minimal function satisfying this inequality, called metric derivative and denoted by . The metric derivative of can be computed for a.e. as
[TABLE]
It is known that (see for example [7]) the length of a curve can be computed as
[TABLE]
In particular, on a length space we have
[TABLE]
where the infimum is taken among all which connect and .
Given , the local Lipschitz constant is defined as
[TABLE]
if is not isolated, [math] otherwise, while the (global) Lipschitz constant is defined as
[TABLE]
If is a length space, we have .
We are not only interested in metric structures, but also in the interaction between metric and measure. For the metric measure space , basic assumptions used in this paper are:
Assumption 2.1*.*
The metric measure space satisfies:
- •
is a complete and separable length space,
- •
is a non-negative Borel measure with respect to and finite on bounded sets,
- •
.
In this paper, we will often assume that the metric measure space satisfies the condition, for some and (when it is denoted by ). The and conditions are refinements of the curvature-dimensions proposed by Lott-Sturm-Villani (see [25] and [30, 31] for ), and Bacher-Sturm (see [9] for ) in order to isolate the non-smooth ‘Riemannian’ structures from the ‘Finslerian’ ones. More precisely, the conditions are obtained by reinforcing the corresponding conditions by adding the requirement that the Sobolev space is a Hilbert space (see the next subsection for more details). It is then clear that the following relations hold
[TABLE]
moreover one has that
[TABLE]
It is known that, for finite , a space satisfies the following properties:
- •
is locally doubling and therefore a locally compact space, [9];
- •
supports a local Poincaré inequality, [29].
For more details about and spaces, we refer to [3, 1, 5, 16].
2.2. Optimal transport and Sobolev functions
The set of Borel probability measures on will be denoted by . We also use to denote the set of measures with finite second moment, i.e. if and for some (and thus every) . For , the evaluation map is given by
[TABLE]
The space is naturally endowed with the quadratic transportation distance defined by:
[TABLE]
where the is taken among all couplings with marginals and , i.e. and where , are the projection maps onto the first and second coordinate respectively. The metric space will be denoted by . Let us recall that the infimum in the Kantorovich problem (2.2) is always attained by an optimal coupling . We denote the set of optimal couplings between and by . Below we recall some fundamental properties of the metric space we will use throughout the paper.
Proposition 2.2** (Geodesics in the Wasserstein space).**
Let be a metric space and fix . Then the curve is a constant speed geodesic connecting and , i.e. it satisfies
[TABLE]
if and only if there exists , called optimal dynamical plan (or simply optimal plan), such that
[TABLE]
*The set of optimal dynamical plans from to is denoted by .
Moreover, if is a geodesic space, then is also geodesic.*
Absolutely continuous curves in are characterized by the following theorem:
Theorem 2.3** (Superposition principle, [24]).**
Let be a complete and separable metric space and let . Then there exists a measure concentrated on such that:
[TABLE]
Moreover, the infimum of the energy among all the satisfying for every is attained by such .
Definition 2.4** (Test plan).**
Let be a metric measure space and . We say that has bounded compression provided there exists such that
[TABLE]
We say that is a test plan if it has bounded compression, is concentrated on and
[TABLE]
The notion of Sobolev function is given in duality with that of test plan:
Definition 2.5** (The Sobolev class ).**
Let be a metric measure space. A Borel function belongs to the Sobolev class (resp. ) provided there exists a non-negative function (resp. ) such that
[TABLE]
In this case, is called a 2-weak upper gradient of , or simply weak upper gradient.
It is known, see e.g. [2], that there exists a minimal function in the -a.e. sense among all the weak upper gradients of . We denote such minimal function by or to emphasize which space we are considering and call it minimal weak upper gradient. Notice that if is Lipschitz, then -a.e., because is a weak upper gradient of .
It is known that the locality holds for , i.e. -a.e. on the set , moreover is a vector space and the inequality
[TABLE]
holds for every and . Moreover, the space is an algebra, with the inequality
[TABLE]
being valid for any .
The Sobolev space , also denoted by for short, is defined as
[TABLE]
and is endowed with the norm
[TABLE]
is always a Banach space, but in general it is not a Hilbert space. is said infinitesimally Hilbertian if is a Hilbert space.
On an infinitesimally Hilbertian space, we have a natural pointwise inner product defined by
[TABLE]
In order to prove the cosine formula we will use properties of harmonic functions in open sets of a m.m. space. Let us define the relevant quantities and recall the properties we will use; for simplicity, as always we assume the space to be proper, complete and separable, and the measure to be finite on bounded sets (this indeed is the geometric case correspoding to spaces, for we will be interested in). For the general case see for instance [6, 17, 19].
Definition 2.6** (Sobolev classes in ).**
Let be a m.m. space and let be an open subset. The space is the space of Borel functions such that for any Lipschitz function such that , where is taken [math] by definition on . Let be the corresponding Sobolev space endowed with the natural norm, and denote by the closure of compactly supported Lipschitz functions on .
Definition 2.7** (Measure valued Laplacian).**
Let be a m.m. space, an open subset and a Borel function. We say that is in the domain of the Laplacian in , and write provided and there exists a locally finite Borel measure on such that for any with compact support contained in it holds
[TABLE]
In this case the measure is unique and we denote it by , or simply . If , we denote its density with respect to by or simply by .
A function is said to be harmonic in , or simply harmonic, if .
For simplicity we state the next proposition for space, though it is valid more generally for doubling spaces supporting a weak-local 1-2 Poincaré inequality (see [6] for details).
Proposition 2.8**.**
Let be a space, for some and , and let be a bounded open set. Then the following properties hold.
- i)
Regularity*. Let be harmonic in . Then admits a continuous representative (actually even locally Lipschitz).*
- ii)
Comparison*. If are such that -a.e. on for some and then -a.e. on .*
- ii)
Existence and uniqueness of harmonic functions*. Assume that and let . Then there exists a unique harmonic function on such that .*
- iv)
Strong maximum principle*. Let be harmonic in and assume that its continuous representative has a maximum at a point . Then is constant on the connected component of containing .*
In order to state the Laplacian Comparison Theorem, let us introduce the coefficients defined by
[TABLE]
Theorem 2.9** (Laplacian comparison, [17]).**
Let be an space for some and . Then
[TABLE]
and
[TABLE]
2.3. Pointed measured Gromov-Hausdorff convergence and convergence of functions
In order to study the convergence of possibly non-compact metric measure spaces, it is useful to fix reference points. We then say that is a pointed metric measure space, p.m.m.s. for short, if is a m.m.s. as before and plays the role of reference point. Recall that, for simplicity, we always assume . We will adopt the following definition of convergence of p.m.m.s. (see [7], [20] and [32]):
Definition 2.10** (Pointed measured Gromov-Hausdorff convergence).**
A sequence is said to converge in the pointed measured Gromov-Hausdorff topology (p-mGH for short) to if there exists a separable metric space and isometric embeddings such that for every and there exists such that for every
[TABLE]
where for every subset , and
[TABLE]
where denotes the set of real valued bounded continuous functions with bounded support in .
Sometimes in the following, for simplicity of notation, we will identify the spaces with their isomorphic copies . It is obvious that this is in fact a notion of convergence for isomorphism classes of p.m.m.s., moreover it is induced by a metric (see e.g. [20] for details).
Next, following [20], we recall various notions of convergence of functions defined on p-mGH converging spaces.
Definition 2.11** (Pointwise convergence of scalar valued functions).**
Let , be a p-mGH converging sequence of p.m.m.s. and let be a sequence of functions. We say that converge pointwise to provided:
[TABLE]
If for any there exists such that for every and every with , we say that uniformly.
Definition 2.12** ( weak and strong convergence).**
Let , be a p-mGH converging sequence of pointed metric measure spaces and let be a sequence of functions.
- •
We say that converges weakly in to provided weakly as Radon measures, i.e.
[TABLE]
and
[TABLE]
- •
We say that converges strongly in to provided it converges weakly in to and moreover
[TABLE]
Definition 2.13** ( weak and strong convergence).**
Let , be a p-mGH converging sequence of pointed metric measure spaces and let be a sequence of functions. We say that converges weakly in to if are -weakly convergent to and
[TABLE]
Strong convergence in is defined by requiring -strong convergence of the functions and that
[TABLE]
The next result proved in [4, Corollary 5.5] (see also [20, Corollary 6.10]) will be useful in the sequel.
Proposition 2.14**.**
Let , be a p-mGH converging sequence of pointed metric measure spaces. If for every one has , with uniformly bounded in , and converges strongly in to , then and converges to strongly in .
2.4. Euclidean tangent cones to spaces
Let us first recall the notion of measured tangents. Let be a m.m.s., and ; we consider the rescaled and normalized p.m.m.s. where the measure is given by
[TABLE]
Then we define:
Definition 2.15** (Tangent cone and regularity).**
Let be a m.m.s. and . A p.m.m.s. is called a tangent to at if there exists a sequence of rescalings so that as in the p-mGH sense. We denote the collection of all the tangents of at by . A point is called regular if the tangent is unique and euclidean, i.e. if , where is the Euclidean distance and is the properly rescaled Lebesgue measure of .
The a.e. regularity was settled for Ricci-limit spaces by Cheeger-Colding [12, 13, 14]; for an -space , it was proved in [21] that for -a.e. there exists a blow-up sequence converging to a Euclidean space. The -a.e. uniqueness of the blow-up limit, together with the rectifiability of an -space, was then established in [27]. More precisely the following holds:
Theorem 2.16** (-a.e. infinitesimal regularity of -spaces).**
Let be an -space for some . Then -a.e. is a regular point, i.e. for -a.e. there exists such that, for any sequence , the rescaled pointed metric measure spaces converge in the p-mGH sense to the pointed Euclidean space .
3. Definition of angle
3.1. Angle between three points
In [26], the second author proposed a notion of angle between three points in a metric space . In general such an angle is non unique, the possible causes of non-uniqueness being a lack of regularity of the distance function (e.g. is in the cut locus of or ) or a lack of infinitesimal strict convexity of the distance function (for more details we refer to [26, Sections 1,2]). For simplicity, here we only treat the case when the angle is unique. Given two points , consider the distance functions
[TABLE]
Definition 3.1**.**
We say that the angle exists if and only if the limit for of the quantity exists. In this case we set
[TABLE]
Note that if is a smooth Riemannian manifold and is not in the cut locus of and , then is the angle based at between and ; in other words is the angle based at “in direction of and ”. As already mentioned, for a general triple in a general metric space the angle may not exist; moreover, even if both and exist they may not be equal in general. On the other hand, such a definition satisfies some natural properties one expects from the geometric picture: the angle is invariant under a constant rescaling of the metric , moreover for any two points the angle always exists and, if is a length space, is equal to [math].
We now discuss an important class of metric measure spaces where the angle exists and is symmetric in an a.e. sense, the so called Lipschitz-infinitesimally Hilbertian spaces.
Definition 3.2**.**
A metric measure space is said to be Lipschitz-infinitesimally Hilbertian if for any pair of Lipschitz functions both the limits for of and exist and are equal for -a.e. , i.e.
[TABLE]
It is clear that if is Lipschitz-infinitesimally Hilbertian then, given , for -a.e. both the angles exist and .
Remark 3.3*.*
The concept of Lipschitz-infinitesimally Hilbertian space was proposed in [26] as a variant of the notion of infinitesimally Hilbertian space introduced in [3, 17], using the language of minimal weak upper gradients; let us mention that Lipschitz-infinitesimally Hilbertian always implies infinitesimally Hilbertian, but the converse is not clear in general. An important class of spaces where also the converse implication holds is the one of locally doubling spaces satisfying a weak Poincaré inequality. Indeed, by a celebrated result of Cheeger [11], we have that for every it holds -a.e., in other words the local Lipschitz constant is equal to the minimal weak upper gradient -a.e. In particular for spaces, the two notions are equivalent. For more details we refer to [26, Remark 3.3].
It follows that -spaces are Lipschitz-infinitesimally Hilbertian, for ; let us recall that the class of -spaces include finite dimensional Alexandrov spaces with curvature bounded below and Ricci limit spaces as remarkable sub-classes.
3.2. Angle between two geodesics
First of all observe that if is a metric space and is a geodesic, then for a.e. ; we will denote such a constant simply by . The next definition is inspired by the De Giorgi’s metric concept of gradient flow [15].
Definition 3.4** (A geodesic representing the gradient of a Lipschitz function).**
Let be a Lipschitz function on . We say that represents at time [math], or represents the gradient of at the point if the following inequality holds
[TABLE]
Notice that the opposite inequality is always true, indeed
[TABLE]
Hence represents at time [math] if and only if the equality holds. Note that, in the case of Riemannian manifolds, represents at time [math] if and only if .
It is easy to check that the geodesic represents the gradient of at if and only if for every the rescaled geodesic defined by , , represents the gradient of the Lipschitz function at . In the next lemma we give a simple but important example of a geodesic representing the gradient of a function.
Lemma 3.5**.**
Let be a metric space, fix and let . If for some there exists a geodesic such that and then represents the gradient of at .
Proof.
For every it holds
[TABLE]
On the other hand, by triangle inequality it is clear that and with an analogous argument as above it is easily checked that actually . Therefore and the claim follows. ∎
We can now define the angle between two geodesics.
Definition 3.6** (Angle between two geodesics).**
Let be a metric space and let be two geodesics with . Let be a Lipschitz function such that represents the gradient of at time [math]. We say that the angle exists if and only if the limit as of exists. In this case we set
[TABLE]
Remark 3.7* (Locality of the angle between two geodesics).*
It is easily seen that the angle between the two geodesics at the point depend just on the germs of the curves at . To see that, fix arbitrary and call the restrictions of to properly rescaled, i.e:
[TABLE]
Of course we still have , and it is readily seen that represents the gradient of . It follows that exists if and only if exists, and in this case it holds
[TABLE]
Remark 3.8* (Dependence on the function ).*
111AM:added this remark
Note also in the generality of metric spaces, the angle as given in Definition 3.6 may depend on the function chosen in (3.5) (for instance this is the case of a tree with a vertex in and two edges made by and ). In case is an -space we will see later in the paper that actually the angle between two geodesics is well defined for -a.e. base point just in terms of the geometric data, so it does not depend on the choice of . In the general case of a metric space, a way to overcome the problem would be to fix a canonical Lipschitz function such that represents at time [math]. In view of Lemma 3.5, a natural choice is to consider . In case is not an space we will tacitly make such a choice so to have a good definition.
The next goal is to relate the angle between three points with the angle between two geodesics, i.e. relate Definitions 3.1 and 3.6.
Theorem 3.9**.**
222AM: this thm and the proof is improved from the last version, where there was problem due to the lack of locality of the angle between three points
Let be a metric space and let satisfy the following assumptions:
- •
the angle exists in the sense of Definition 3.1,
- •
there exists geodesic from to and from to respectively.
Then the angle exists in the sense of Definition 3.6 and
[TABLE]
Note that if is a geodesic Lipschitz-infinitesimally Hilbertian space then for every given the two assumptions of Theorem 3.9 are satisfied for -a.e. . This is in particular the case for spaces (see Remark 3.3).
Proof.
Without loss of generality we can assume that , otherwise just exchange the role of and in the arguments below. Let and . Recall from Lemma 3.5 that represents at in the sense of Definition 3.4, i.e.
[TABLE]
On the other hand
[TABLE]
Subtracting (3.7) from (3.8) yields
[TABLE]
If , dividing both sides by and letting we get
[TABLE]
where in the last identity we used the assumption that exists. Analogously, if , dividing both sides by and letting we get
[TABLE]
The combination of the last two inequalities gives the existence of the limit for of and, more precisely,
[TABLE]
Since by Lemma 3.5 we know that represents the gradient of at in the sense of Definition 3.4, we get that the rescaled geodesic defined by , , represents the gradient of at .
Therefore exists in the sense of Definition 3.6 and ; recalling the locality of the angle between geodesics (see Remark 3.7), we conclude that exists in the sense of Definition 3.6 and . ∎
3.3. Angles in Wasserstein spaces
In the Wasserstein space, we have the notion of “Plans representing gradients” which is similar to the one of “geodesic representing the gradient” above.
Definition 3.10** (Plans representing gradients, see [17]).**
Let be a metric measure space, and be a test plan. We say that represents the gradient of if
[TABLE]
Let be with uniformly bounded densities, be its lifting given by Theorem 2.3, and let . In case is infinitesimally Hilbertian, it is proved in [18, Theorem 4.6] that represents the gradient of if and only if
[TABLE]
If (3.10) holds, we also say that the velocity field of at time [math] is .
Combing the above technical tools with ideas from Otto’s calculus [28], we can define the angle between two geodesics in .
Definition 3.11** (Angle between curves in ).**
Let be an infinitesimally Hilbertian metric measure space, let be with bounded compression, and such that . Assume there exist lifting test plans of and representing the gradients of and respectively, for some . Then the angle between and at is defined by
[TABLE]
The same definition makes sense if provided have uniformly bounded supports.
From the formula (3.10), we can see that the value of the angle does not depend on the choice of , but just on , .
Remark 3.12* (Locality of the angle in the Wasserstein space).*
The angle depends just on the germs of the curves and at ; i.e., given , called for all the restrictions of to and respectively, it holds . Indeed let , lift of the curve , be a test plan representing the gradient of ; fix and let for every be the restriction of the curve to ; called the lift of , it is easily seen that represents the gradient of . The claim follows.
Thanks to the locality expressed in Remark 3.12, given two curves such that they are of bounded compression once restricted to for some , we can define the angle between them as the angle between their restrictions to . This will be always tacitly assumed throughout the paper.
Let us briefly discuss the particular but important case when and are -geodesics in a general m.m.s. . If is a -geodesic with bounded compression then any lift of is a test plan and moreover is an optimal dynamical plan, i.e. . Moreover, as a consequence of the Metric Brenier Theorem proved in [2] (see also [18, Theorem 5.2] for the present formulation), if has bounded compression and is a Kantorovich potential from to , then any lift of represents the gradient of . Therefore, specializing Definition 3.11 to this case we get the following notion.
Definition 3.13** (Angle between geodesics in ).**
Let be an infinitesimally Hilbertian metric measure space, let be -geodesics with bounded compression, and such that . Assume there exist Kantorovich potentials from to and from to respectively. Then the angle between and at is defined by
[TABLE]
The same definition makes sense if provided have uniformly bounded supports.
Note that, thanks to Otto calculus and (3.10), Definition 3.13 is the analog for geometry of the angle between two geodesics in a general metric space in the sense of Definition 3.6.
3.4. The case of spaces
In Theorem 3.9 we related the angle between three points with the angle between two geodesics, i.e. we related Definitions 3.1 and 3.6. Now, adding a curvature assumption on the space, we wish to relate Definition 3.13 with Definition 3.1 and Definition 3.6, i.e. the angle between two geodesics in with the angle between three points and the angle between two geodesics of . To this aim the next lemma will be useful.
Lemma 3.14**.**
Let be an space, and let be locally Lipschitz functions on . Then the functions
[TABLE]
are well defined at every and it holds
[TABLE]
Proof.
From the definition of local Lipschitz constant we know that the function is convex for any . Consider the function
[TABLE]
and observe that is non-decreasing on and for any fixed . Hence and are well-defined for any point as
[TABLE]
Since is a metric measure space, it holds a local Poincaré inequality and it is locally doubling. Then, from [11, Theorem 6.1], we know for -a.e. . The definition of infinitesimal Hilbertian space and of , then gives
[TABLE]
and
[TABLE]
Hence
[TABLE]
In particular, we infer -a.e.. ∎
In the next result we relate Definition 3.13 with Definition 3.1. Before stating it, let us recall [22, Theorem 1.1] that if is an m.m.s., with , then there exists a unique geodesic connecting and ; let us mention that the same result holds more generally for essentially non-branching m.m.s satisfying the weaker condition [10].
Proposition 3.15**.**
Let be an m.m.s. and fix . For every and let and let be the unique -geodesics from to and from to respectively. Then
[TABLE]
Proof.
Calling as usual , Lemma 3.14 implies
[TABLE]
On the other hand, it is easily seen that are Kantorovich potentials from to and from to respectively. Moreover the geodesics have uniformly bounded supports, and bounded compression once restricted to for every , see for instance [22, Corollary 1.7]. Then, Definition 3.13 yields
[TABLE]
The combination of the two formulas gives the claim. ∎
Remark 3.16*.*
For uniformity with the rest of the paper we decided to state Proposition 3.15 for spaces, but using the results of [10] the same conclusion holds for essentially non-branching Lipschitz-infinitesimally Hilbertian spaces satisfying .
In the next result we relate Definition 3.6 with the optimal transport picture.
Proposition 3.17**.**
Assume that is an metric measure space. Let and be -geodesics with bounded compression and with ; let be corresponding lifts. Then, for , we can find with , such that for -a.e. there exist unique geodesics with , and the angle exists according to the Definition 3.6. Moreover
[TABLE]
where is any locally Lipschitz Kantorovich potential from to .
Proof.
From [3, 18] we know that any lift of represents the gradient of , for , i.e:
[TABLE]
From [17, Proposition 3.11] we then get for :
[TABLE]
In other words, for -a.e. , we have that represents at , .
For any , consider the function and observe that
[TABLE]
The difference between (3.14) and (3.13), for , gives
[TABLE]
Multiplying by both sides of (3.15) yields
[TABLE]
Letting and using Lemma 3.14, we infer
[TABLE]
Following verbatim the same arguments after (3.13), but now for , gives
[TABLE]
Since from [22, Theorem 3.4] we can find with , such that for -a.e. there exists unique geodesics with , it follows that
[TABLE]
Recalling from (3.13) that -a.e. represents the gradient of at , we get that for -a.e. the geodesic represents the gradient of at . Therefore (3.16) proves that for -a.e. the angle exists according to Definition 3.6 and coincides with \arccos\big{(}{\langle}\nabla\varphi_{2},\nabla\varphi_{1}{\rangle}(x)\big{)}. With the same arguments, just exchanging with , we get that also exists for -a.e. , and that the identities (3.12) hold. ∎
4. The cosine formula for angles in spaces
The goal of this section is to prove Theorem 4.4, stating that the cosine formula holds for the angle between two geodesics in an space. The first lemma states the almost everywhere uniqueness and extendability of geodesics in spaces; this fact is already present in the literature under slightly different formulations so we just briefly sketch the proof.
Lemma 4.1**.**
Let be an space for some , and fix . Then for -a.e. there exist unique geodesics such that
- •
, , ,
- •
both and are extendable to geodesics and having as interior point; in other words there exist and such that and
Proof.
Step 1. , -a.e. is an interior point of a geodesic with end point at .
Fix and . Consider
[TABLE]
Analyzing the optimal transport from to by following verbatim the proof of [21, Lemma 3.1] (i.e. use Jensen’s inequality and the convexity property of the entropy granted by the curvature condition), we get that for -a.e. there exists a geodesic such that and , for some . The claim then follows by the arbitrariness of .
Step 2. , -a.e. there exists a unique geodesic from to .
The uniqueness of geodesics connecting a fixed and -a.e. is a consequence of [22, Theorem 3.5] applied to the optimal transportation from the measures , above.
Step 3. Applying steps 1 and 2 to and , since the union of two negligible sets is still negligible, the thesis follows. ∎
The next lemma will be useful to get good estimates on harmonic approximations of distance functions.
Lemma 4.2**.**
Let be a unit ball in an metric measure space , . Then there exists a function with such that
[TABLE]
where is a constant which depends only on and .
Proof.
Since is a metric measure space, it satisfies a local (1-2)-Poincaré inequality and it is locally doubling. It is also known [3, Remark 6.9 and Theorem 6.10] that the metric is induced by the Dirichlet form . Therefore the standing assumptions of [8] are fulfilled and from [8, Corollary 1.2] we know that for any , , there exists a function such that
[TABLE]
for any . In other words, we know and
[TABLE]
Furthermore, from [8, Theorem 4.1] we know
[TABLE]
where only depends on the constants in the Poincaré inequality and in the doubling condition. In our case, only depends on and .
Now, choosing on , we get that satisfies the thesis with . ∎
Using Lemma 4.2, in the next proposition we prove a key estimate in order to establish the cosine formula for angles.
Proposition 4.3**.**
Let be an metric measure space, for some , and fix . Let , such that , and . We denote and . Assume that there exists a function satisfying for fixed , such that for any .
Then there exists a harmonic approximation of with the following properties:
- (1)
, with , 2. (2)
it holds
[TABLE]
where satisfies for fixed .
Proof.
From Proposition 2.8 we know there exists satisfying (1) of the thesis. Similarly, we can find a harmonic approximation of .
We are then left to show the validity of the estimate (4.1). To this aim, let be given by Lemma 4.2, so that
[TABLE]
where depends only on and in particular is independent of .
From Laplacian Comparison Theorem 2.9 we know that and
[TABLE]
and
[TABLE]
for some suitable satisfying for fixed . Then we have
[TABLE]
Applying the comparison statement of Proposition 2.8 to we get that
[TABLE]
Analogously, applying the comparison statement of Proposition 2.8 to we get
[TABLE]
By assumption, we know there exists a function satisfying for fixed , such that for any . Using maximum principle of Proposition 2.8, we know
[TABLE]
for any . The combination of the last three estimates gives
[TABLE]
Putting together (4.4) and (4.5), we get
[TABLE]
Next, write for short. Recalling that , combining (4.2) with (4.6) and using that in order to integrate by parts, we obtain
[TABLE]
where we used that, since , it holds
[TABLE]
Summing up (4.6) and (4.7) we get (4.1) by renaming with the quantity 2\big{(}2C(K,N)\,\Psi(R|K,N)+\,\Phi(R|K,N)\big{)}\,\Psi(R|K,N)+2\,C(K,N)\,\Psi(R|K,N)+\Phi(R|K,N). ∎
Theorem 4.4**.**
Let be an space for some , and fix . Then for -a.e. let be the unique geodesics from to and from to given by Lemma 4.1. We may also assume that the tangent cone at is unique and isomorphic as m.m. space to , for some . Let be any sequence, be the limit points of under the rescalings which converge to in p-mGH sense. Then
[TABLE]
Proof.
Step 1. Fix . Combining Theorem 2.16, Remark 3.3, Theorem 3.9 and Lemma 4.1 we get that for -a.e.
- •
we can find unique geodesics such that , , , and both are extendable beyond in the sense of Lemma 4.1, so we can assume that could be extended to respectively, for some ,
- •
both the angles and exist in the sense of Definitions 3.1, 3.6 respectively, and ,
- •
is a Lebesgue point for so that
[TABLE]
- •
the tangent to at is unique and euclidean.
From the locality of the angle (see Remark 3.7) we know that
[TABLE]
where for all .
Let be any sequence and let be the corresponding sequence of rescaled spaces. Since by assumption is regular, we know that p-mGH converge to for some . Since by assumption both and are extendable beyond , they converge in p-GH sense to half lines in such that and both are extendable to full lines of . We parametrize such half lines on such that for every one has that are the limit points of , respectively. Denote by be the limit points of . By the uniqueness of the tangent space, the parametrized half lines and the points do not depend on the choice of the rescaling sequence .
Let be the half lines in antipodal to respectively; in other words and are straight lines in intersecting at . We parametrize on such that , for all .
Step 2. We claim that
[TABLE]
Since the first identity is true by construction, and the last is trivially true because the ambient space is , it is enough to show that . Given any sequence of rescalings , let and define
[TABLE]
Set also to be the Busemann functions associated to , i.e.
[TABLE]
Since by construction we know that in p-GH sense, and for every , it follows that
[TABLE]
More strongly, since are all Lipschitz with unit Lipschitz constant, by an Arzelá-Ascoli procedure (see for instance in [27, Proposition 2.12]) we get that the convergences are uniform on bounded subsets, in the sense of Definition 2.11. In particular, since the measures are converging weakly to we get that
[TABLE]
Define analogously to (4.12)-(4.13):
[TABLE]
With analogous arguments as above we get
[TABLE]
Since we are working in the euclidean space , it is not difficult to see that and . Note that such equalities holds more generally in manifolds with non-negative Ricci curvature: the argument, used in the proof of the Cheeger-Gromoll Splitting Theorem, goes via maximum principle; here in any case one can argue more directly by using the geometry of the euclidean space. It follows that and uniformly on bounded sets in the sense of Definition 2.11. Hence there exists a function satisfying for fixed , such that and for any .
Using Proposition 4.3, for every we can construct harmonic approximations of , respectively, in the unit ball of the space . Since , and the spaces are RCD, so in particular RCD for large enough, we infer that
[TABLE]
The combination of (4.15), (4.16) and (4.17) yields
[TABLE]
Since by construction , by Proposition 2.14 we get that
[TABLE]
But then the gradient estimates in (4.16)-(4.17) give that
[TABLE]
In particular, for every we have
[TABLE]
We now analyze the two sides of (4.19). Recalling (4.9), from the very definitions of and of it follows that
[TABLE]
On the other hand, since are the Busemann functions of the lines in it is readily seen that
[TABLE]
Putting together (4.19), (4.20) and (4.21) finally yields
[TABLE]
as desired.
Step 3. We claim that
[TABLE]
To this aim, first of all observe that the cosine formula in ensures that
[TABLE]
Let now be any sequence and set . Define the rescaled spaces as above, with . Notice first of all that the p-GH convergence of to ensures that
[TABLE]
It follows that
[TABLE]
Since the sequence was arbitrary, (4.25) implies (4.22).
The thesis then follows by combining (4.11) and (4.22).
∎
Remark 4.5*.*
The cosine formula in ensures that
[TABLE]
It is natural to ask if the same formula holds in the non-smooth case. This remains an open problem even for Ricci limit spaces, so a fortiori in spaces.
Here let us briefly mention that with analogous arguments as above one can show the weaker statement
[TABLE]
To this aim let be any two sequences. Up to subsequences, we may assume that for all it holds either or . Without loss of generality we may assume the first case. Up to further subsequences we may also assume that has a limit as . Let and define the rescaled spaces as above, with . Calling , , the p-GH convergence of to ensures
[TABLE]
Then we have
[TABLE]
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