The Hyperbolic Metric

Let $X$ be a Riemann surface with a constant negative curvature. A metric on $X$ is a way of assigning a distance between any two points on the surface. The hyperbolic metric is a specific type of metric that is conformally equivalent to a given metric on $X$, but has constant sectional curvature equal to $-1$.

The hyperbolic metric can be constructed as follows. Let $ds^2$ be a metric on $X$. For any point $p$ in $X$, let $B(p, r)$ be the open ball of radius $r$ centered at $p$ with respect to $ds^2$. The area of $B(p, r)$ is given by:

$$\text{Area}(B(p, r)) = 2\pi(1 - e^{-2r})$$

where $e$ is the base of the natural logarithm. The hyperbolic metric on $X$ is then defined by the formula:

$$ds_h^2 = \frac{4}{(1 - |z|^2)^2}|dz|^2$$

where $z$ is a local complex coordinate on $X$ and $|dz|$ is the Euclidean norm of the differential $dz$ with respect to the coordinate system.

The hyperbolic metric has several remarkable properties. One of the most important is that it is invariant under conformal transformations, meaning that it is preserved when the surface is stretched or compressed in a conformal manner. This property makes the hyperbolic metric useful for studying the conformal geometry of Riemann surfaces.

Another important property of the hyperbolic metric is that it satisfies the Gauss-Bonnet theorem, which relates the curvature of the metric to the topology of the surface. In particular, if $X$ is a closed Riemann surface (i.e., compact without boundary), then the total curvature of the hyperbolic metric is proportional to the Euler characteristic of $X$.

The hyperbolic metric also has applications in various areas of mathematics, including topology, complex analysis, and number theory. For example, the uniformization theorem states that any simply connected Riemann surface is conformally equivalent to either the complex plane, the unit disk with the hyperbolic metric, or the Riemann sphere with the spherical metric. This theorem has important implications for the study of Teichmüller theory and the moduli space of Riemann surfaces.

Relation to the Lemniscate of Bernoulli

there is a connection between the hyperbolic metric and the lemniscate of Bernoulli, which is a curve in the complex plane named after the Swiss mathematician Jacob Bernoulli.

The lemniscate of Bernoulli is defined by the equation:

$$|\mathbf{z}|^2 = \frac{1}{2}(a^2 + b^2) - \frac{1}{2}\sqrt{(a^2 + b^2)^2 - 4a^2b^2}$$

where $\mathbf{z} = x + iy$ is a complex number, and $a$ and $b$ are real constants.

The hyperbolic metric on the unit disk can be expressed in terms of the Poincaré disk model, which is a way of representing the hyperbolic plane as a disk in the complex plane. In this model, the hyperbolic distance between two points is given by:

$$d(\mathbf{z}_1,\mathbf{z}_2) = \operatorname{arcosh}\left(1 + \frac{2|\mathbf{z}_1 - \mathbf{z}_2|^2}{(1 - |\mathbf{z}_1|^2)(1 - |\mathbf{z}_2|^2)}\right)$$

where $\operatorname{arcosh}$ is the inverse hyperbolic cosine function.

If we define $a = \operatorname{sech}(r)$ and $b = \tanh(r)$, then the lemniscate of Bernoulli can be expressed as:

$$|\mathbf{z}|^2 = \frac{1}{2}(a^2 + b^2) - \frac{1}{2}\sqrt{(a^2 + b^2)^2 - 4a^2b^2} = \frac{1}{1 - |\mathbf{z}|^2}$$

which is the equation for the unit circle in the Poincaré disk model. Thus, the lemniscate of Bernoulli is related to the hyperbolic metric on the unit disk in a subtle way.

The connection between the lemniscate of Bernoulli and the hyperbolic metric is also related to the theory of elliptic functions, which are functions that are periodic in the complex plane with respect to a lattice of periods. The lemniscate of Bernoulli can be expressed in terms of the Weierstrass elliptic function, which is a fundamental tool in the study of elliptic curves and modular forms.

Subtleties

The connection between the lemniscate of Bernoulli and the hyperbolic metric on the unit disk is subtle because it is not immediately obvious from the formulas. While it is true that the lemniscate of Bernoulli can be expressed in terms of the hyperbolic metric, the relationship between the two concepts is not straightforward.

One reason for this is that the lemniscate of Bernoulli is defined in terms of algebraic equations, while the hyperbolic metric is defined in terms of differential geometry. The two fields have different approaches and techniques, and it is not always easy to see how they are related.

Another reason is that the connection between the lemniscate of Bernoulli and the hyperbolic metric is more conceptual than visual. While the lemniscate of Bernoulli is a well-known curve with a geometric interpretation, the hyperbolic metric is a more abstract concept that is defined in terms of distances and curvatures. The relationship between the two is more mathematical than geometric, and it requires some background in complex analysis and differential geometry to appreciate fully.

That being said, the connection between the lemniscate of Bernoulli and the hyperbolic metric is an important one, and it highlights the deep connections between different areas of mathematics. By exploring these connections, we can gain a deeper understanding of the underlying structures and principles that govern the mathematical universe.

A Curve That Vanishes on the Lemniscate of Bernoulli

The hyperbolic metric can be written in terms of the Poincaré disk model as:

$$ds_h^2 = \frac{4}{(1 - |z|^2)^2}|dz|^2$$

and if we define $a = \operatorname{sech}(r)$ and $b = \tanh(r)$, then the equation of the lemniscate of Bernoulli can be written as:

$$|\mathbf{z}|^2 = \frac{1}{2}(a^2 + b^2) - \frac{1}{2}\sqrt{(a^2 + b^2)^2 - 4a^2b^2} = \frac{1}{1 - |\mathbf{z}|^2}$$

where $a = \operatorname{sech}(r)$ and $b = \tanh(r)$. To see why, note that the real part of the $T$ function is given by:

$$\operatorname{Re}(T(t)) = \frac{t^4 - 6t^2 + 1}{(t^2 + 1)^2}$$

The zeros of the real part of $T$ are the values of $t$ that satisfy $t^4 - 6t^2 + 1 = 0$. These values of $t$ correspond to the intersection points of the curve $y^2 = x^2 - x^4$ with the real line in the complex plane, which is the lemniscate of Bernoulli. Therefore, the zeros of the real part of the $T$ function in the upper half-plane model correspond to the points on the lemniscate of Bernoulli.

Summary

In summary, the $T$ function is related to the Weierstrass function by a transformation that maps the Poincaré half-plane model to the upper half-plane model of the hyperbolic plane. The Weierstrass function is a periodic function with poles at the lattice points in the complex plane, while the $T$ function is a meromorphic function with zeros on the real axis of the upper half-plane model. The transformation between the two functions allows us to study the geometry of the hyperbolic plane in terms of the properties of the $T$ function and the lemniscate of Bernoulli.

The zeros of the real part of the $T$ function correspond to the points on the lemniscate of Bernoulli, which is a curve with interesting geometric properties. This provides a way to study the lemniscate of Bernoulli and its properties in terms of the hyperbolic geometry of the unit disk. Conversely, the properties of the lemniscate of Bernoulli can be used to derive properties of the $T$ function and the Weierstrass function.

Overall, the connection between the $T$ function, the Weierstrass function, and the lemniscate of Bernoulli highlights the deep connections between different areas of mathematics. By exploring these connections, we can gain a deeper understanding of the underlying structures and principles that govern the mathematical universe.