Can we detect Gaussian curvature by counting paths and measuring their length?

The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image of the initial paths in $\mathbb{R}^2$ under the given parameterization. Moreover, we prove that this integral is given by the average of the lengths of the external paths times the measure of the set of paths if and only if the surface has Gaussian curvature equal to zero.


Introduction
In the theory of smooth surfaces the Gaussian curvature plays a fundamental role as a tool to measure how these objects bend in Euclidean space. It is an intrinsic function depending only on distances that are measured on the surface, but not in how the surface is isometrically embedded in R 3 . This result is the famous Gauss' Theorema Egregium.
Can we detect this curvature by counting paths and measuring their length in a given surface? To fix ideas, let us consider a connected surface S and two given points p, q ∈ S. We will not consider all possible paths on S from p to q, but only simple ones as we shall explain. To settle down the problem, we start with the simplest case, i.e., the Euclidean plane R 2 , of constant curvature equal to zero, when equipped with Cartesian coordinates (x, y) and the Riemannian metric dx 2 + dy 2 . The idea relies on looking only at paths built with straight lines parallel to the axes, and only moving forward. In other words, we concatenate a finite number of pieces of the flows ((x, y), s) → (x + s, y), ((x, y), s) → (x, y + s), corresponding to the orthogonal vector fields ∂ x and ∂ y coming from the given coordinates. Then we can reach q = (x 1 , y 1 ) from p = (x 0 , y 0 ) if and only if x 1 ≥ x 0 , y 1 ≥ y 0 and the time we need to do this is given by t = x 1 − x 0 + y 1 − y 0 . In this way we have established the set Γ p,q (t) of all admissible paths. The first question is: how can we count or give a measure to this set? In this particular case, every element of Γ p,q (t) has the same length, i.e., t. Thus, independently of how we define the measure value m(Γ p,q (t)), when we add all these lengths, the result will be t · m(Γ p,q (t)). The conclusion is the following: in curvature zero the "sum" of all these lengths is equal to the average of the lengths of the external paths times the "number" of paths. The external paths are the one from (x 0 , y 0 ) to (x 1 , y 0 ) and then to (x 1 , y 1 ), and the other from (x 0 , y 0 ) to (x 0 , y 1 ) and then to (x 1 , y 1 ).
The aim of this note is to show that we can effectively define m(Γ p,q (t)) and the "sum", or better, the integral of the length over Γ p,q (t), to prove that the phenomenon explained in the previous paragraph only occurs in the case of surfaces of curvature zero. We will transfer the situation of the Euclidean plane to S by using a local parameterization Ψ : U ⊆ R 2 → S containing p = Ψ(x 0 , y 0 ) and q = Ψ(x 1 , y 1 ). Our calculations work for metrics of the form where h, f : R → R are strictly increasing smooth functions. We remark that, when |f ′′ (s)| < h ′ (s), surfaces of revolution realize these spaces (see Section 4 for details). Since the classical surfaces of constant curvature, i.e, the plane R 2 , the sphere S 2 and the Poincaré half plane H = R >0 × R, can be realized locally as open revolution surfaces in R 3 , our result is valid for these surfaces as well.
We will see that a good definition for the measure of the set of paths Γ p,q (t) explained above is given by a n (t − a) n (n + 1)!n! , denotes the so called continuous binomial coefficient defined in [4]. The name and suggestive notation come from the analogy with the binomial coefficient n+m n which counts the paths in the lattice N 2 ⊂ R 2 , from (0, 0) to (m, n) ∈ N 2 . Then, our main result is an explicit formula for the integral of the length associated to the Riemannian metric (1), which is the content of: Theorem 1.1. The integral of the length over all admissible paths Γ p,q (t) from p = Ψ(x 0 , y 0 ) to q = Ψ(x 1 , y 1 ) in time t > 0 is given by where t = x 1 − x 0 + y 1 − y 0 and F (τ, s) := τ +s s . In particular, the integral is given by the measure of This formula gives us a new insight on how the curvature of a surface affects the length of paths defined over it. We will prove it by studying a quite simple definition of a path integral and looking at the geometric information it contains.
It is worth remarking that several approaches to path integrals exist in the literature. The most famous ones are Feynman and Wiener integrals. Feynman integrals, although not fully formalized from a mathematical point of view, are used in Quantum Mechanics to capture physical information, see [6,5]. The Wiener integrals are defined on sets of continuous paths in Euclidean space and model Brownian motion. However, the differential paths have Wiener measure zero, see e.g., [5, Thm. 1.1], and hence do not have a notion of length. It could be interesting to see if the integral of length makes some sense in any of the mathematical formulations of Feynman integrals.

Paths as simplexes and their measure
Coming back to the case of the plane R 2 , a path in Γ p,q (t) is determined, first, by the order in which we glue a finite number of segments given by the flows of ∂ x and ∂ y , and second, by the length of these segments: for the horizontal (resp. vertical) part, the length is x 1 − x 0 , (resp. y 1 − y 0 ). Such order can be represented by a finite sequence c of the numbers 1 and 2, where 1 represents an horizontal segment and 2 represents a vertical one, i.e., by an element of the set In this way, we can write as a disjoint union, where each Γ c p,q (t) denotes the set of paths built with order c. If c ∈ C(n), a path γ ∈ Γ c p,q (t) corresponds to a vector s = (s 0 , s 1 , . . . , s n ) ∈ R n+1 >0 , where s j is the length of the jth segment of γ. We can distinguish four cases: In all cases, Γ c p,q (t) is given, up to a permutation of the variables, by the Cartesian product of two simplexes. Thus we can reduce the problem to measure usual simplexes in Euclidean space.
By using the parameterization ϕ τ1 , formula that can be interpreted as a multiplicative counting principle in our setting. Moreover, if we fix 0 ≤ m < n, ∆ τ n can be written as the disjoint union 0≤s≤τ ∆ s m × ∆ τ −s n−1−m . The measure we have introduced is compatible with this decomposition in the following sense: Note we have used here the elementary formula for the Beta function for the case II.2. Finally, the measure of Γ p,q (t) is defined as the sum of the measures of its parts, in agreement with the definition of the binomial coefficient t x1−x0 given in the Introduction.
Remark 2.1. Our approach to associate this measure is based on the work [3]. There the authors work more generally with a manifold M and a finite number of vector fields X 1 , . . . , X k defined over it. In this framework, if p, q ∈ M and t > 0 are given, the set of admissible paths Γ p,q (t) consists of piece-wise smooth paths from p to q formed by concatenations of the flows φ j of the X j using a total time t. Here there is also a decomposition of Γ p,q (t) analogous to (4) and if c = (c 0 .c 1 , . . . , c n ) is a given order, Γ c p,q (t) can be embedded in ∆ t n .

On the continuous binomial coefficients
The motivation to introduce the power series t a , which converges for every t, a ∈ C, came from measure or "count" elements of Γ p,q (t), thus we regard it as a continuous analogue (not extension) of the classical binomial coefficients. For extensions to complex variables, it is enough to write the usual binomial coefficients in terms of the Gamma function. For a more detailed study of this extension to real variables, see e.g. [7].
To show the strong parallel with the discrete binomial coefficients, we highlight the following two properties τ + s s = τ + s τ , = n+m m , respectively, and D n f = f (n + 1) − f (n) is a difference operator. There are more analogies, for instance, continuous versions of Chu-Vandermonde's formula, see [8].
To perform some calculations we will need later, we can use the Bessel-Clifford function of the first kind [2, p. 358] which are defined by the power series and where Γ denotes the classical gamma function. From the very definition, we note that dCν dz = C ν+1 and also (8) zC ν+2 (z) + (ν + 1)C ν+1 (z) = C ν (z).
where the powers of z assume their principal value (here f (z) ∼ g(z) means that lim z→∞ f (z)/g(z) = 1 in the corresponding domain). Then, it follows from equation (9) that
If we choose h(s) = s and f (s) = − cos(s), 0 < s < π, the previous conditions are fulfilled and we obtain a typical local parameterization of the sphere S 2 . On the other hand, by taking h(y) = f (y) = ln(y), the metric would be y −2 (dx 2 + dy 2 ) which is the metric of the Poincaré plane H. Thus, if we put and we change 1/s = sin(t), we recover the usual parameterization of the pseudosphere obtained by rotating the tractrix This is a model of an open surface of constant Gaussian curvature equal to −1, see e.g., [1, p. 198].
Naturally, if h(s) = f (s) = 1 we obtain the cylinder, which has curvature identically zero. Yet another model for curvature zero is the punctured plane in polar coordinates, i.e., η(s) = (s, 0, 0), s > 0, h(s) = s, f (s) = s 2 /2, and the metric ds 2 + s 2 dφ 2 . In these cases Theorem 1.1 shows that the integral of the length is given by the measure of Γ p,q (t) times the average of the lengths of the paths with configurations (1, 2) and (2, 1), since f is quadratic.
We are now in position to define and compute the integral of the length over Γ p,q (t), and thus to prove Theorem 1.1. The first observation is that the length of the horizontal part of any path in Γ p,q (t) is actually constant, and it is given by h(x 1 ) − h(x 0 ). Then, it is enough to calculate the integral of the length of the vertical parts. For this, we use the decomposition (4) and distinguish the four cases I.1, I.2, II.1 and II.2 as explained in Section 2. To simplify notations we will write a = x 1 − x 0 and t − a = y 1 − y 0 . f ′ (x 0 + l j+1 ) (l j+1 − l j ).
When we integrate this function over W a m−1 × W t−a m−1 , we find that it is given by where I m−1 (a, t − a) is equal to f ′ (x 0 + l j+1 ) (l j+1 − l j )dl 1 · · · dl m−1 dl 1 · · · dl m−1 To solve this recurrence and the ones that will appear in the other cases, we can use the following