2.4 Parametrized surfaces

Parametrized surfaces are a key concept in differential geometry, allowing us to study two-dimensional surfaces in three-dimensional space. They provide a mathematical framework for describing and analyzing the shape, curvature, and other properties of surfaces.

This topic covers the definition of parametrized surfaces, local parametrizations, Riemannian metrics, isometries, Gaussian curvature, geodesics, minimal surfaces, and the Gauss-Bonnet theorem. These concepts form the foundation for understanding the geometry of curved spaces.

Definition of parametrized surfaces

• A parametrized surface is a smooth mapping $X: U \to \mathbb{R}^3$ from an open set $U \subset \mathbb{R}^2$ to three-dimensional Euclidean space $\mathbb{R}^3$
• Parametrized surfaces provide a way to describe and study two-dimensional surfaces embedded in three-dimensional space, which is a fundamental concept in differential geometry
• The mapping $X$ assigns to each point $(u,v) \in U$ a point $X(u,v) = (x(u,v), y(u,v), z(u,v))$ on the surface in $\mathbb{R}^3$

Local parametrizations

Coordinate patches

• A coordinate patch is a pair $(U, X)$, where $U \subset \mathbb{R}^2$ is an open set and $X: U \to \mathbb{R}^3$ is a smooth mapping
• Coordinate patches provide a local description of a surface, allowing for the study of local properties such as tangent planes, curvature, and geodesics
• The mapping $X$ is often required to be a homeomorphism onto its image, ensuring that the patch is a faithful representation of the surface locally

Transition maps

• Transition maps, also known as change of coordinate maps, describe how different coordinate patches of a surface are related to each other
• Given two overlapping coordinate patches $(U, X)$ and $(V, Y)$, the transition map $\varphi: X(U \cap V) \to Y(U \cap V)$ is defined by $\varphi = Y \circ X^{-1}$
• Transition maps are required to be smooth (differentiable) to ensure that the surface is well-defined and has consistent properties across different patches

Atlas for a surface

• An atlas for a surface is a collection of coordinate patches $\{(U_i, X_i)\}_{i \in I}$ that cover the entire surface, such that any two overlapping patches are related by a smooth transition map
• The existence of an atlas ensures that the surface is a smooth manifold, allowing for the application of differential geometric techniques
• The study of atlases and their properties is central to the field of differential topology, which is closely related to differential geometry

Riemannian metrics on surfaces

First fundamental form

• The first fundamental form, denoted as $I$, is a quadratic form that measures the infinitesimal distance between points on a surface
• In local coordinates $(u,v)$, the first fundamental form is given by $I = E du^2 + 2F du dv + G dv^2$, where $E, F, G$ are the coefficients of the first fundamental form
• The coefficients $E, F, G$ are determined by the inner products of the partial derivatives of the parametrization: $E = \langle X_u, X_u \rangle, F = \langle X_u, X_v \rangle, G = \langle X_v, X_v \rangle$

Induced metric from R^3

• The induced metric on a surface is the Riemannian metric obtained by restricting the Euclidean metric of the ambient space $\mathbb{R}^3$ to the surface
• Given a parametrized surface $X: U \to \mathbb{R}^3$, the induced metric is the first fundamental form determined by the coefficients $E, F, G$ as described above
• The induced metric allows for the measurement of lengths, angles, and areas on the surface, as well as the computation of geometric quantities such as curvature and geodesics

Isometries of surfaces

Isometric surfaces

• Two surfaces $S_1$ and $S_2$ are said to be isometric if there exists a diffeomorphism $f: S_1 \to S_2$ that preserves the Riemannian metric, i.e., the pullback of the metric on $S_2$ by $f$ equals the metric on $S_1$
• Isometric surfaces have the same intrinsic geometry, meaning that they have the same Gaussian curvature, geodesics, and other metric-dependent properties
• Examples of isometric surfaces include a plane and a cylinder, or a catenoid and a helicoid

Local isometries

• A local isometry between two surfaces $S_1$ and $S_2$ is a smooth mapping $f: S_1 \to S_2$ that preserves the Riemannian metric in a neighborhood of each point
• Local isometries preserve the intrinsic geometry of the surfaces locally, but may not be globally one-to-one or onto
• The study of local isometries is important in understanding the local behavior of surfaces and their relationships to each other

Gaussian curvature of surfaces

Definition of Gaussian curvature

• The Gaussian curvature $K$ of a surface at a point $p$ is the product of the principal curvatures $\kappa_1$ and $\kappa_2$ at that point, i.e., $K = \kappa_1 \kappa_2$
• Principal curvatures are the eigenvalues of the shape operator (or Weingarten map), which describes how the surface bends in different directions at a given point
• Gaussian curvature is an intrinsic property of the surface, meaning that it depends only on the first fundamental form and not on the embedding of the surface in $\mathbb{R}^3$

Gaussian curvature in local coordinates

• In local coordinates $(u,v)$, the Gaussian curvature can be expressed in terms of the coefficients of the first and second fundamental forms
• The second fundamental form, denoted as $II$, measures how the surface bends in $\mathbb{R}^3$ and is given by $II = L du^2 + 2M du dv + N dv^2$, where $L, M, N$ are the coefficients of the second fundamental form
• The Gaussian curvature is then given by $K = \frac{LN-M^2}{EG-F^2}$, where $E, F, G$ are the coefficients of the first fundamental form

Theorema Egregium

• The Theorema Egregium, or "Remarkable Theorem," states that the Gaussian curvature of a surface is invariant under local isometries
• This means that if two surfaces are locally isometric, they must have the same Gaussian curvature at corresponding points
• The Theorema Egregium highlights the importance of Gaussian curvature as an intrinsic property of surfaces and has far-reaching consequences in the study of geometry and topology

Geodesics on surfaces

Definition of geodesics

• A geodesic on a surface is a curve that locally minimizes the distance between any two nearby points on the curve
• Geodesics are the generalization of straight lines in Euclidean space to curved surfaces
• On a parametrized surface, geodesics are characterized by a system of second-order differential equations called the geodesic equations

Geodesic equations

• The geodesic equations are a system of second-order differential equations that describe the path of a geodesic on a surface in local coordinates
• In terms of the Christoffel symbols $\Gamma_{ij}^k$, which are determined by the coefficients of the first fundamental form, the geodesic equations are given by $\ddot{u}^k + \Gamma_{ij}^k \dot{u}^i \dot{u}^j = 0$, where $u^k$ are the coordinates of the geodesic and dots represent derivatives with respect to the curve parameter
• Solving the geodesic equations allows for the computation of geodesics on a given surface and the study of their properties

Clairaut's relation

• Clairaut's relation is a first integral of the geodesic equations for surfaces of revolution, providing a conserved quantity along geodesics on such surfaces
• For a surface of revolution with parametrization $X(u,v) = (f(u)\cos v, f(u)\sin v, g(u))$, Clairaut's relation states that $f(u)^2 \dot{v} = c$, where $c$ is a constant determined by the initial conditions of the geodesic
• Clairaut's relation simplifies the study of geodesics on surfaces of revolution and allows for the classification of different types of geodesic behavior

Minimal surfaces

Definition of minimal surfaces

• A minimal surface is a surface that locally minimizes its area among all surfaces with the same boundary
• Equivalently, a surface is minimal if its mean curvature, given by $H = \frac{\kappa_1 + \kappa_2}{2}$, vanishes identically
• Examples of minimal surfaces include the plane, the catenoid, and the helicoid

Isothermal coordinates

• Isothermal coordinates are a special parametrization of a surface in which the coefficients of the first fundamental form satisfy $E = G$ and $F = 0$
• In isothermal coordinates, the first fundamental form takes the form $I = \lambda^2(du^2 + dv^2)$, where $\lambda$ is a positive function called the conformal factor
• The existence of isothermal coordinates is guaranteed by the uniformization theorem, and they are particularly useful in the study of minimal surfaces and conformal geometry

Weierstrass-Enneper parameterization

• The Weierstrass-Enneper parameterization is a method for constructing minimal surfaces using complex analysis
• Given two holomorphic functions $\phi(z)$ and $\psi(z)$ satisfying certain conditions, the Weierstrass-Enneper parameterization defines a minimal surface $X(u,v) = \text{Re} \int_{z_0}^z (\phi(z), i\phi(z), \psi(z)) dz$, where $z = u + iv$ is a complex variable
• This parameterization provides a powerful tool for generating and studying minimal surfaces, and has led to the discovery of many interesting examples

Gauss-Bonnet theorem

Statement of Gauss-Bonnet theorem

• The Gauss-Bonnet theorem relates the total Gaussian curvature of a closed surface to its Euler characteristic, a topological invariant
• For a closed surface $S$, the theorem states that $\int_S K dA + \int_{\partial S} \kappa_g ds = 2\pi \chi(S)$, where $K$ is the Gaussian curvature, $dA$ is the area element, $\kappa_g$ is the geodesic curvature of the boundary (if any), $ds$ is the line element along the boundary, and $\chi(S)$ is the Euler characteristic of the surface
• The Gauss-Bonnet theorem is a deep result that connects the intrinsic geometry of a surface to its topology, and has numerous applications in mathematics and physics

Applications of Gauss-Bonnet

• The Gauss-Bonnet theorem can be used to compute the Euler characteristic of a surface, which provides information about its topology (genus, number of holes, etc.)
• In physics, the theorem plays a crucial role in the study of general relativity and the geometry of spacetime, as it relates curvature to topological properties
• The Gauss-Bonnet theorem also has applications in the study of geometric flows, such as the Ricci flow, which has been used to prove the Poincaré conjecture and other important results in geometry and topology

Classification of surfaces

• The Gauss-Bonnet theorem, combined with other results in differential geometry and topology, allows for the classification of closed surfaces based on their Euler characteristic
• Surfaces with Euler characteristic $\chi = 2$ are topologically equivalent to a sphere, those with $\chi = 0$ are equivalent to a torus (or a connected sum of tori), and those with $\chi < 0$ are equivalent to a connected sum of projective planes
• This classification provides a powerful framework for understanding the global structure of surfaces and has far-reaching implications in various areas of mathematics, including algebraic geometry, complex analysis, and topology