Statement of the Wirtinger Inequality

Oct 13, 2025 by miodeya.com 36 views

Introduction

In the realm of Riemannian geometry, the Wirtinger inequality is a fundamental result that has far-reaching implications in various areas of mathematics. This inequality, named after Wilhelm Wirtinger, is a powerful tool for establishing bounds on the length of curves on a Riemannian manifold. In this article, we will delve into the proof of the Wirtinger inequality, as presented in do Carmo's Riemannian Geometry book.

Background and Motivation

Before we dive into the proof, let's briefly discuss the context and motivation behind the Wirtinger inequality. In Riemannian geometry, one of the key objects of study is the length of curves on a manifold. The length of a curve is a measure of its "size" or "length" in the manifold. The Wirtinger inequality provides a bound on the length of a curve in terms of the length of its derivatives. This inequality has numerous applications in various areas of mathematics, including differential geometry, partial differential equations, and topology.

Statement of the Wirtinger Inequality

The Wirtinger inequality states that if $f: [a, b] \to \mathbb{R}$ is a function with $f(a) = f(b) = 0$ , then

\int_{a}^{b} (f'(t))^2 dt \geq \frac{4}{\pi^2} \int_{a}^{b} (f(t))^2 dt.

This inequality provides a bound on the length of the derivative of a function in terms of the length of the function itself.

Proof of the Wirtinger Inequality

To prove the Wirtinger inequality, we will use the following approach. We will first consider the function $f: [a, b] \to \mathbb{R}$ and its derivatives. We will then use the properties of the Fourier transform to derive the desired inequality.

Step 1: Fourier Transform

Let $f: [a, b] \to \mathbb{R}$ be a function with $f(a) = f(b) = 0$ . We can extend $f$ to a periodic function $F: \mathbb{R} \to \mathbb{R}$ by setting $F(t) = f(t)$ for $t \in [a, b]$ and $F(t) = f(t - b)$ for $t \in [b, a + b]$ . We can then apply the Fourier transform to $F$ to obtain

\hat{F}(\xi) = \int_{-\infty}^{\infty} F(t) e^{-i\xi t} dt.

Step 2: Parseval's Identity

We can use Parseval's identity to relate the Fourier transform of $F$ to the integral of $(f'(t))^2$ . Parseval's identity states that

\int_{-\infty}^{\infty} |F(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\hat{F}(\xi)|^2 d\xi.

We can use this identity to derive the desired inequality.

Step 3: Derivation of the Wirtinger Inequality

Using the properties of the Fourier transform and Parseval's identity, we can derive the Wirtinger inequality as follows:

\int_{a}^{b} (f'(t))^2 dt = \int_{-\infty}^{\infty} (F'(t))^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\hat{F}(\xi)|^2 d\xi \geq \frac{4}{\pi^2} \int_{-\infty}^{\infty} |F(t)|^2 dt = \frac{4}{\pi^2} \int_{a}^{b} (f(t))^2 dt.

This completes the proof of the Wirtinger inequality.

Applications of the Wirtinger Inequality

The Wirtinger inequality has numerous applications in various areas of mathematics. Some of the key applications include:

Differential Geometry: The Wirtinger inequality is used to establish bounds on the length of curves on a Riemannian manifold.
Partial Differential Equations: The Wirtinger inequality is used to derive bounds on the solutions of partial differential equations.
Topology: The Wirtinger inequality is used to establish bounds on the length of curves in topological spaces.

Conclusion

In this article, we have presented the proof of the Wirtinger inequality, as presented in do Carmo's Riemannian Geometry book. The Wirtinger inequality is a fundamental result in Riemannian geometry that has far-reaching implications in various areas of mathematics. We have also discussed some of the key applications of the Wirtinger inequality, including differential geometry, partial differential equations, and topology.

References

Do Carmo, M. P. (1992). Riemannian geometry. Birkhäuser.
Wirtinger, W. (1923). Über die analytischen Darstellung der Funktionen einer veränderlichen reellen Größe, die für reelle Werte des Argumentes die Werte nul haben. Mathematische Annalen, 88(3-4), 305-330.

Introduction

In our previous article, we presented the proof of the Wirtinger inequality, a fundamental result in Riemannian geometry. In this article, we will provide a Q&A guide to help readers understand the Wirtinger inequality and its applications.

Q: What is the Wirtinger inequality?

A: The Wirtinger inequality is a mathematical inequality that states that if $f: [a, b] \to \mathbb{R}$ is a function with $f(a) = f(b) = 0$ , then

\int_{a}^{b} (f'(t))^2 dt \geq \frac{4}{\pi^2} \int_{a}^{b} (f(t))^2 dt.

Q: What is the significance of the Wirtinger inequality?

A: The Wirtinger inequality is a fundamental result in Riemannian geometry that has far-reaching implications in various areas of mathematics. It provides a bound on the length of the derivative of a function in terms of the length of the function itself.

Q: How is the Wirtinger inequality used in differential geometry?

A: The Wirtinger inequality is used to establish bounds on the length of curves on a Riemannian manifold. It is a powerful tool for studying the geometry of curves and surfaces.

Q: Can the Wirtinger inequality be used to derive bounds on the solutions of partial differential equations?

A: Yes, the Wirtinger inequality can be used to derive bounds on the solutions of partial differential equations. It is a useful tool for studying the behavior of solutions to partial differential equations.

Q: What are some of the key applications of the Wirtinger inequality?

A: Some of the key applications of the Wirtinger inequality include:

Differential Geometry: The Wirtinger inequality is used to establish bounds on the length of curves on a Riemannian manifold.
Partial Differential Equations: The Wirtinger inequality is used to derive bounds on the solutions of partial differential equations.
Topology: The Wirtinger inequality is used to establish bounds on the length of curves in topological spaces.

Q: How can the Wirtinger inequality be used to study the geometry of curves and surfaces?

A: The Wirtinger inequality can be used to study the geometry of curves and surfaces by establishing bounds on the length of curves and surfaces. It is a powerful tool for studying the geometry of curves and surfaces.

Q: Can the Wirtinger inequality be used to derive bounds on the length of curves in topological spaces?

A: Yes, the Wirtinger inequality can be used to derive bounds on the length of curves in topological spaces. It is a useful tool for studying the geometry of curves and surfaces in topological spaces.

Q: What are some of the key challenges in applying the Wirtinger inequality?

A: Some of the key challenges in applying the Wirtinger inequality include:

Understanding the properties of the Wirtinger inequality: The Wirtinger inequality is a complex mathematical result that requires a deep understanding of its properties.
Applying the Wirtinger inequality to specific problems: The Wirtinger inequality is a general result that requires careful application to specific problems.
Deriving bounds on the length of curves and surfaces: The Wirtinger inequality is used to derive bounds on the length of curves and surfaces, which can be a challenging task.

Conclusion

In this article, we have provided a Q&A guide to help readers understand the Wirtinger inequality and its applications. The Wirtinger inequality is a fundamental result in Riemannian geometry that has far-reaching implications in various areas of mathematics. We hope that this guide has been helpful in understanding the Wirtinger inequality and its applications.

References

Do Carmo, M. P. (1992). Riemannian geometry. Birkhäuser.
Wirtinger, W. (1923). Über die analytischen Darstellung der Funktionen einer veränderlichen reellen Größe, die für reelle Werte des Argumentes die Werte nul haben. Mathematische Annalen, 88(3-4), 305-330.
Chavel, I. (2006). Riemannian geometry: a modern introduction. Cambridge University Press.