12.3 Non-Euclidean Geometries in hyperbolic manifolds and topology

Hyperbolic manifolds are mind-bending spaces with constant negative curvature. They're like funhouse mirrors for geometry, where parallel lines act weird and triangles have funky angles. These spaces break the rules we learned in high school math.

Non-Euclidean geometries open up a whole new world of mathematical possibilities. They help us understand knots, 3D shapes, and even the fabric of spacetime. It's like putting on special glasses that let us see hidden patterns in the universe.

Hyperbolic Manifolds and Non-Euclidean Geometries

Properties of hyperbolic manifolds

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• Hyperbolic manifolds
  • Manifolds possessing constant negative curvature throughout their structure
  • Locally modeled on hyperbolic space $\mathbb{H}^n$ which has a unique geometric structure differing from Euclidean space
  • Geodesics (shortest paths between points) diverge exponentially as they move away from each other (compared to straight lines in Euclidean space)
• Connection to non-Euclidean geometries
  • Hyperbolic geometry serves as a prime example of a non-Euclidean geometry, contrasting with traditional Euclidean geometry
  • Violates Euclid's parallel postulate which states that given a line and a point not on the line, there is exactly one line through the point parallel to the given line
  • In hyperbolic geometry, given a line $l$ and a point $P$ not on $l$, there exist infinitely many lines passing through $P$ that do not intersect $l$, showcasing the abundance of non-intersecting lines
• Models of hyperbolic geometry
  • Poincaré disk model
    • Points represented by points inside a unit disk (circle of radius 1 centered at the origin)
    • Lines represented by circular arcs perpendicular to the boundary of the disk (unit circle)
  • Upper half-plane model
    • Points represented by points in the upper half-plane $\{(x,y) \in \mathbb{R}^2 : y > 0\}$ (all points above the x-axis)
    • Lines represented by semicircles perpendicular to the $x$-axis or vertical lines extending upwards from the x-axis

Applications in knot theory

• Knots and links
  • Knots mathematically defined as embeddings of the circle $S^1$ into the 3-sphere $S^3$ (3-dimensional space compactified by adding a point at infinity)
  • Links defined as disjoint unions of knots, consisting of multiple non-intersecting knots
  • Hyperbolic structures on knot and link complements
    • Many knots and links have complements (the space obtained by removing the knot or link from $S^3$) that admit complete hyperbolic metrics
    • The hyperbolic structure provides geometric invariants of the knot or link, allowing for their classification and study
• Three-dimensional manifolds
  • Thurston's geometrization conjecture
    • Every closed, orientable, prime 3-manifold can be decomposed into pieces, each admitting one of eight geometric structures (including hyperbolic geometry)
    • Hyperbolic geometry plays a significant role in the classification and understanding of 3-manifolds
  • Hyperbolic volume as a topological invariant
    • The hyperbolic volume of a hyperbolic 3-manifold remains constant under continuous deformations (homeomorphisms)
    • Can be used to distinguish non-homeomorphic 3-manifolds, serving as a powerful tool in the study of 3-dimensional topology

Non-Euclidean Geometries, Topology, and Symmetry

Non-Euclidean geometries for topological spaces

• Geometric structures on manifolds
  • A manifold can admit different geometric structures depending on its properties and characteristics
  • The type of geometry (Euclidean, hyperbolic, spherical, etc.) provides valuable insights into the manifold's shape, curvature, and overall structure
• Curvature and topology
  • The curvature of a manifold is intrinsically linked to its topological properties, revealing fundamental aspects of its shape and structure
  • Gauss-Bonnet theorem establishes a deep connection between the total curvature of a surface and its Euler characteristic (a topological invariant)
• Geometrization of 3-manifolds
  • Thurston's geometrization conjecture provides a framework for classifying 3-manifolds using geometric structures
  • Non-Euclidean geometries, particularly hyperbolic geometry, play a pivotal role in this classification, showcasing their importance in the study of 3-dimensional topology

Connections to group theory and symmetries

• Isometry groups
  • The isometry group of a geometric space consists of all transformations that preserve distances between points
  • Isometry groups of non-Euclidean spaces
    • Hyperbolic space: $\mathrm{PSL}(2,\mathbb{R})$ for $\mathbb{H}^2$, $\mathrm{PSO}(n,1)$ for $\mathbb{H}^n$ (groups of matrices preserving the hyperbolic metric)
    • Spherical space: $\mathrm{SO}(n+1)$ for $S^n$ (special orthogonal group acting on the n-sphere)
• Discrete subgroups and symmetries
  • Discrete subgroups of isometry groups give rise to symmetric patterns and tilings on surfaces and in higher dimensions
  • Examples: hyperbolic tilings (regular tessellations of the hyperbolic plane), Platonic solids (highly symmetric 3D shapes), crystallographic groups (symmetry groups of crystals)
• Applications in physics
  • Non-Euclidean geometries in general relativity
    • Spacetime can exhibit non-Euclidean geometry due to the presence of matter and energy, as described by Einstein's theory of general relativity
  • Discrete symmetries in quantum mechanics
    • Symmetry groups play a fundamental role in describing quantum mechanical systems and their properties
    • Examples: point groups (symmetry groups of molecules), space groups (symmetry groups of crystals), and their representations in terms of matrices or operators