📐Metric Differential Geometry Unit 2 – Curves and surfaces

Key Concepts and Definitions

• Metric spaces generalize the notion of distance, allowing for a rigorous study of geometric properties
• A metric on a set $X$ is a function $d: X \times X \to \mathbb{R}$ satisfying non-negativity, symmetry, and the triangle inequality
  • Non-negativity: $d(x, y) \geq 0$ for all $x, y \in X$, and $d(x, y) = 0$ if and only if $x = y$
  • Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$
  • Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in X$
• Riemannian manifolds are smooth manifolds equipped with a Riemannian metric, enabling the study of geometric properties using differential calculus
• The Riemannian metric is a smooth, positive-definite, symmetric (0,2)-tensor field that assigns an inner product to each tangent space
• Isometries are distance-preserving maps between metric spaces, playing a crucial role in understanding the symmetries and rigidity of geometric structures
• The Levi-Civita connection is a unique, torsion-free, metric-compatible connection on a Riemannian manifold, essential for defining parallel transport and curvature

Curves in Metric Spaces

• Curves in metric spaces are continuous maps from an interval $I \subset \mathbb{R}$ to a metric space $(X, d)$
• The length of a curve $\gamma: [a, b] \to X$ is defined as the supremum of the sums of distances between points on the curve over all partitions of $[a, b]$
  • Formally, $L(\gamma) = \sup \left\{ \sum_{i=1}^{n} d(\gamma(t_{i-1}), \gamma(t_i)) : a = t_0 < t_1 < \cdots < t_n = b \right\}$
• Rectifiable curves are curves with finite length, forming an important class of curves in metric geometry
• Geodesics are locally length-minimizing curves, generalizing the notion of straight lines in Euclidean space
  • In a Riemannian manifold, geodesics are characterized by the vanishing of the covariant derivative of their velocity vector field
• The exponential map $\exp_p: T_pM \to M$ on a Riemannian manifold maps tangent vectors to the endpoints of geodesics starting at $p$, providing a local diffeomorphism around the origin of $T_pM$

Surface Theory Basics

• Surfaces are two-dimensional topological manifolds, locally homeomorphic to open subsets of $\mathbb{R}^2$
• Smooth surfaces are equipped with a differential structure, allowing for the use of differential calculus in their study
• The first fundamental form $I$ is a symmetric (0,2)-tensor field on a surface that encodes the Riemannian metric induced by the ambient Euclidean space
  • In local coordinates $(u, v)$, $I = E du^2 + 2F du dv + G dv^2$, where $E, F, G$ are the coefficients of the first fundamental form
• The second fundamental form $II$ is a symmetric (0,2)-tensor field that measures the extrinsic curvature of a surface in the ambient space
  • In local coordinates, $II = L du^2 + 2M du dv + N dv^2$, where $L, M, N$ are the coefficients of the second fundamental form
• The Gauss map $N: S \to \mathbb{S}^2$ assigns to each point on a surface its unit normal vector, providing insight into the local geometry of the surface
• The Weingarten map $L_p: T_pS \to T_pS$ is a self-adjoint linear operator that measures the change in the Gauss map along tangent vectors, related to the second fundamental form by $II(X, Y) = \langle L_p(X), Y \rangle$

Curvature and Geodesics

• Gaussian curvature $K$ is an intrinsic measure of curvature on a surface, defined as the determinant of the Weingarten map
  • $K = \det(L_p) = \frac{LN - M^2}{EG - F^2}$, where $E, F, G$ and $L, M, N$ are the coefficients of the first and second fundamental forms, respectively
• Mean curvature $H$ is the average of the principal curvatures, measuring the extrinsic curvature of a surface
  • $H = \frac{1}{2} \text{tr}(L_p) = \frac{EN - 2FM + GL}{2(EG - F^2)}$
• Minimal surfaces are surfaces with vanishing mean curvature, characterized by locally area-minimizing properties
  • Examples include the catenoid, helicoid, and Enneper's surface
• Geodesics on surfaces are curves that locally minimize the distance between points, generalizing the notion of straight lines
  • Geodesics are characterized by the vanishing of their geodesic curvature, which measures the deviation of the curve from being a geodesic
• The Gauss-Bonnet theorem relates the total Gaussian curvature of a compact surface to its Euler characteristic, providing a deep connection between curvature and topology
  • $\int_S K dA + \int_{\partial S} k_g ds = 2\pi \chi(S)$, where $k_g$ is the geodesic curvature of the boundary and $\chi(S)$ is the Euler characteristic

Metric Properties of Surfaces

• The Riemannian distance function $d: S \times S \to \mathbb{R}$ on a surface measures the infimum of the lengths of all curves connecting two points
  • $d(p, q) = \inf \left\{ L(\gamma) : \gamma \text{ is a curve from } p \text{ to } q \right\}$
• The diameter of a surface is the maximum distance between any two points, providing a global measure of the surface's size
  • $\text{diam}(S) = \sup \left\{ d(p, q) : p, q \in S \right\}$
• The injectivity radius at a point $p \in S$ is the largest radius for which the exponential map $\exp_p$ is a diffeomorphism, measuring the size of the largest geodesic ball around $p$
• Geodesic completeness is a property of surfaces where every geodesic can be extended indefinitely, ensuring the existence of geodesics between any two points
• The Hopf-Rinow theorem characterizes geodesic completeness in terms of the completeness of the metric space $(S, d)$, connecting the metric and geodesic properties of surfaces

Applications in Differential Geometry

• The Uniformization Theorem states that every simply connected Riemann surface is conformally equivalent to the Euclidean plane, the hyperbolic plane, or the Riemann sphere
  • This result has important applications in the study of complex analysis and the geometry of surfaces
• The Riemann Mapping Theorem ensures the existence of a conformal map between any simply connected proper subset of the complex plane and the unit disk
• Minimal surfaces have applications in architecture and engineering, as they optimize material usage while maintaining structural integrity
  • Examples include the Munich Olympic Stadium and the Denver International Airport roof
• Riemannian geometry has applications in general relativity, where spacetime is modeled as a 4-dimensional Lorentzian manifold
  • The curvature of spacetime is related to the presence of matter and energy through Einstein's field equations
• Geodesics play a crucial role in the study of dynamical systems and the behavior of particles in curved spaces, such as the motion of satellites around Earth

Advanced Topics and Extensions

• Riemann surfaces are one-dimensional complex manifolds, providing a rich interplay between complex analysis, algebraic geometry, and differential geometry
• Kähler manifolds are complex manifolds with a compatible Riemannian metric, leading to a harmonious interaction between complex and Riemannian geometry
  • Kähler geometry has applications in mathematical physics, particularly in the study of supersymmetry and string theory
• The Atiyah-Singer Index Theorem relates the analytical properties of differential operators to the topological properties of the underlying manifold, bridging the gap between analysis and geometry
• The study of curvature flows, such as the Ricci flow and the mean curvature flow, has led to significant advances in the understanding of the topology and geometry of manifolds
  • Perelman's proof of the Poincaré conjecture relies on the analysis of the Ricci flow on 3-manifolds
• Higher-dimensional generalizations of surface theory, such as the study of hypersurfaces and submanifolds, lead to a rich theory with applications in various branches of mathematics and physics

Problem-Solving Strategies

• Visualize the problem using geometric intuition, drawing pictures and diagrams to gain insight into the structure of the surface or curve
• Utilize local coordinates and computations to simplify the problem and express geometric quantities in terms of the coefficients of the first and second fundamental forms
• Exploit symmetries and isometries to simplify the problem, reducing the complexity of the calculations involved
• Apply the appropriate theorems and results from differential geometry, such as the Gauss-Bonnet theorem or the Hopf-Rinow theorem, to solve the problem
• Break down the problem into smaller, more manageable parts, solving each component separately and then combining the results
• Consider special cases or examples to gain intuition about the general problem, using the insights gained to guide the problem-solving process
• Verify the solution by checking that it satisfies the given conditions and is consistent with known results or theorems in differential geometry
• Explore alternative approaches and methods, as there may be multiple ways to solve a given problem in differential geometry