Question

Given a compact metric space \(\mathrm{R}\), suppose two points A and B in \(\mathrm{R}\) can be joined by a continuous curve of finite length. Prove that among all such curves, there is a curve of least length. Comment. Even in the case where \(R\) is a "smooth" (i.e., sufficiently differentiable) closed surface in Euclidean 3-space, this result is not amenable to the methods of elementary differential geometry, which ordinarily deals only with the case of "neighboring" points A and B.

Short answer

In compact metric space, a curve of minimal length between two points exists due to compactness and lower semi-continuity of length.

Step-by-step solution

Define a Continuous Curve

To begin solving the problem, consider a continuous curve \( \gamma(t): [0, 1] \rightarrow R \) that joins points A and B, with \( \gamma(0) = A \) and \( \gamma(1) = B \). This curve represents a path from A to B inside the compact metric space \( R \).

Consider the Set of All Curve Lengths

Examine the set \( S \) consisting of the lengths of all continuous curves in \( R \) that join A and B. This is defined as \( S = \{ L(\gamma) \mid \gamma \text{ is a continuous path from A to B} \} \).

Show Set S is Non-Empty, Bounded Below

Since a continuous curve between A and B exists, at least one element exists in \( S \), making it non-empty. The lengths are positive real numbers, providing a lower bound of 0 for \( S \).

Apply Compactness of Metric Space

Since \( R \) is a compact metric space, every sequence of curves \( \gamma_n \) within it has a subsequence that converges to some continuous curve \( \gamma^* \) in \( R \). Compactness ensures that this limit curve is also valid between A and B.

Use the Invariance of Length Limiting Process

The length function \( L(\gamma) \) is lower semi-continuous, meaning it satisfies the condition \( L(\gamma^*) \leq \liminf_{n \to \infty} L(\gamma_n) \). Therefore, the curve \( \gamma^* \) can potentially achieve the infimum of the curve lengths joining A and B.

Confirm Existence of Minimal Curve

Since \( S \) is non-empty and has a lower bound, and because lower semi-continuity allows \( \gamma^* \) to achieve the least length, there exists a curve of smallest length in the metric space \( R \).

Key concepts

Continuous Curve

A continuous curve is essentially a path that smoothly connects two points within a particular space. Imagine standing at point A and walking toward point B without ever lifting your feet off the ground or jumping over any point along the way. Mathematically, a continuous curve from point A to point B in a space \( R \) can be represented as a function \( \gamma(t) : [0, 1] \to R \), where \( \gamma(0) = A \) and \( \gamma(1) = B \).
This notation simplifies our understanding of a path by offering a formal and clean way to express how a journey from one point to another proceeds without interruptions. Thus, a continuous curve ensures that for every small change in \( t \), there is a correspondingly small alteration in the position within space \( R \). This property is essential when addressing problems involving compact metric spaces and analyzing curves of finite and least length.

Finite Length

A curve's finite length is tied to the idea of measuring how long a path extends between two points in space. Unlike a never-ending line, a finite-length curve has a clear start and finish. For instance, consider traveling from point A to point B along a hilly trail; the total distance you cover is akin to the finite length of this path. In technical terms, if \( L(\gamma) \) denotes the length of a curve \( \gamma \), it is finite if it represents a distinct, bounded numeric value rather than extending indefinitely.
The importance of finite lengths emerges when attempting to compare different paths between the same endpoints and ultimately finding the path with the least length or shortest journey. Finite lengths make the endeavor to find minimal curves feasible because they allow for clear, quantifiable, and comparable measurements.

Curve of Least Length

The curve of least length is the shortest possible path connecting two given points, A and B, within a space. When focusing on compact metric spaces, continuous curves of finite length can vary in length even when connecting the same endpoints. Hence, there's always a quest to find that optimal path — the curve of the least length.
By considering all possible curves between A and B in the set \( S = \{ L(\gamma) \mid \gamma \text{ is a continuous path from A to B} \} \), one can determine the infimum or the greatest lower bound of these lengths. Using mathematical tools such as lower semi-continuity, we validate the existence of such a minimal curve. Essentially, despite the presence of numerous paths, compactness and mathematical structure ensure at least one path emerges as the least lengthy among them.

Compactness

Compactness is a key property in mathematics, reminiscent of a perfectly sealed container where everything inside is nicely packed and closed. In the context of metric spaces, a compact space is one where every sequence of points or curves has a subsequence that converges to a point within the same space. This property plays a crucial role when dealing with continuous curves and finite lengths.
When applying compactness to solve the problem of finding a curve of least length, it's this inherent completeness that ensures convergence. For any sequence of potential curves between points A and B, compactness guarantees that there exists a converging subsequence leading to a well-defined, continuous, and finite-length path within the space \( R \). Thus, compactness acts as a bridge ensuring not only the existence but the attainability of the curve of least length.