📏Geometric Measure Theory Unit 9 – Curvature Measures & Gauss-Bonnet Theorem

Key Concepts and Definitions

• Curvature measures quantify the geometric properties and shape of surfaces and manifolds
• Gaussian curvature $K$ is the product of the principal curvatures $k_1$ and $k_2$ at a point on a surface
  • For a sphere of radius $r$, $K = \frac{1}{r^2}$ at every point
  • Positive curvature (sphere), negative curvature (hyperbolic paraboloid), and zero curvature (plane) are determined by the sign of $K$
• Mean curvature $H$ is the average of the principal curvatures $H = \frac{k_1 + k_2}{2}$
• Geodesic curvature $k_g$ measures the deviation of a curve from being a geodesic on a surface
• Euler characteristic $\chi$ is a topological invariant that describes the shape of a surface
  • For a compact, connected surface $M$, $\chi(M) = 2 - 2g$, where $g$ is the genus (number of holes) of the surface
• Riemannian metric $g$ is a smooth, positive-definite, symmetric bilinear form that defines the inner product on tangent spaces

Curvature in Differential Geometry

• Curvature is a fundamental concept in differential geometry that describes how a surface or manifold deviates from being flat
• Principal curvatures $k_1$ and $k_2$ are the eigenvalues of the shape operator (second fundamental form) at a point
  • The shape operator $S_p: T_pM \to T_pM$ maps tangent vectors to their normal components
• Gaussian curvature is intrinsic, meaning it can be determined solely from the Riemannian metric without embedding the surface in a higher-dimensional space
• Surfaces with constant Gaussian curvature include spheres ($K > 0$), planes ($K = 0$), and hyperbolic surfaces ($K < 0$)
• Gaussian curvature is related to the Riemann curvature tensor $R$, which measures the non-commutativity of covariant derivatives
• Sectional curvature $K(u, v)$ is the Gaussian curvature of the surface spanned by two orthonormal tangent vectors $u$ and $v$

Types of Curvature Measures

• Scalar curvature $S$ is the trace of the Ricci curvature tensor $S = g^{ij}R_{ij}$
  • It provides an average measure of curvature at a point
• Ricci curvature $Ric(u, v) = \text{tr}(w \mapsto R(u, w)v)$ is a quadratic form that measures the average sectional curvature in the direction of a tangent vector
• Mean curvature vector $\vec{H}$ is the trace of the second fundamental form $\vec{H} = \frac{1}{2}g^{ij}h_{ij}\vec{n}$, where $\vec{n}$ is the unit normal vector
  • It points in the direction of the surface's greatest bending
• Total curvature $\int_M KdA$ is the integral of Gaussian curvature over the entire surface
• Geodesic curvature measures how a curve on a surface deviates from being a geodesic (shortest path)
  • For a unit-speed curve $\gamma(s)$, $k_g(s) = \langle \nabla_{\dot{\gamma}}\dot{\gamma}, J\dot{\gamma} \rangle$, where $J$ is the $\frac{\pi}{2}$ rotation in the tangent plane

Gauss-Bonnet Theorem: Statement and Significance

• The Gauss-Bonnet theorem relates the total Gaussian curvature of a compact, oriented, Riemannian 2-manifold $M$ to its Euler characteristic $\chi(M)$
  • $\int_M KdA + \int_{\partial M} k_gds = 2\pi\chi(M)$
  • $dA$ is the area element, and $ds$ is the line element along the boundary $\partial M$
• The theorem establishes a deep connection between the geometry (curvature) and topology (Euler characteristic) of a surface
• It implies that the total Gaussian curvature is a topological invariant, independent of the specific Riemannian metric
• For closed surfaces (no boundary), the theorem simplifies to $\int_M KdA = 2\pi\chi(M)$
  • Example: For a sphere $S^2$, $\chi(S^2) = 2$, so $\int_{S^2} KdA = 4\pi$
• The Gauss-Bonnet theorem has generalizations to higher-dimensional manifolds and singular spaces

Proof Techniques and Applications

• The classical proof of the Gauss-Bonnet theorem relies on triangulating the surface and applying the local Gauss-Bonnet formula to each triangle
  • The local formula states that $\int_T KdA + \sum_{i=1}^3 \int_{e_i} k_gds + \sum_{i=1}^3 \theta_i = 2\pi$, where $T$ is a geodesic triangle, $e_i$ are its edges, and $\theta_i$ are its interior angles
• Modern proofs utilize differential forms and Stokes' theorem
  • The Gaussian curvature 2-form $\Omega = KdA$ and the geodesic curvature 1-form $\omega = k_gds$ satisfy $d\omega = \Omega$
  • Stokes' theorem then yields $\int_M \Omega + \int_{\partial M} \omega = 0$, which is equivalent to the Gauss-Bonnet formula
• The Gauss-Bonnet theorem has applications in geometry, topology, and physics
  • It can be used to compute the Euler characteristic of surfaces and classify them based on their total curvature
  • In general relativity, the theorem relates the curvature of spacetime to its topological properties
• The Chern-Gauss-Bonnet theorem generalizes the Gauss-Bonnet theorem to even-dimensional Riemannian manifolds using the Pfaffian of the curvature form

Historical Context and Development

• Carl Friedrich Gauss first discovered the connection between curvature and topology in his 1827 paper "Disquisitiones generales circa superficies curvas"
  • He derived the local Gauss-Bonnet formula for geodesic triangles on surfaces
• Pierre Ossian Bonnet independently proved the global theorem for closed surfaces in 1848
• The Gauss-Bonnet theorem was a significant milestone in the development of differential geometry and topology
  • It showed that geometric properties (curvature) can have topological implications (Euler characteristic)
• The theorem has been generalized and extended in various directions
  • The Chern-Gauss-Bonnet theorem (1944) for even-dimensional Riemannian manifolds
  • The Atiyah-Singer index theorem (1963) relating the index of an elliptic differential operator to topological invariants
• The Gauss-Bonnet theorem has inspired research in geometric analysis, geometric topology, and mathematical physics

Connections to Other Areas of Mathematics

• The Gauss-Bonnet theorem bridges differential geometry, topology, and analysis
• It is closely related to the Poincaré-Hopf theorem, which relates the Euler characteristic to the indices of zeros of a vector field
  • For a compact, oriented manifold $M$ and a vector field $X$ with isolated zeros, $\chi(M) = \sum_{p \in \text{Zeros}(X)} \text{ind}_p(X)$
• The theorem has analogues in complex geometry, such as the Riemann-Roch theorem for Riemann surfaces and the Hirzebruch-Riemann-Roch theorem for complex manifolds
• In geometric topology, the Gauss-Bonnet theorem is used to study the geometry and topology of surfaces and 3-manifolds
  • The Gauss-Bonnet formula for closed surfaces is a key ingredient in the classification of surfaces
• The theorem has applications in general relativity and the study of gravitational instantons
  • The Gauss-Bonnet-Chern action in 4D is a topological invariant related to the Euler characteristic and signature of the spacetime manifold

Problem-Solving Strategies and Examples

• When applying the Gauss-Bonnet theorem, identify the surface or manifold, its boundary (if any), and its Euler characteristic
• Compute the Gaussian curvature and geodesic curvature using the given Riemannian metric or parametrization
  • Example: For a surface $z = f(x, y)$, the Gaussian curvature is $K = \frac{f_{xx}f_{yy} - f_{xy}^2}{(1 + f_x^2 + f_y^2)^2}$
• Use symmetry and special properties of the surface to simplify calculations
  • Example: For surfaces of revolution, the Gaussian curvature can be expressed in terms of the profile curve
• Subdivide the surface into regions where the curvatures are easier to compute or integrate
  • Example: For a cube, compute the total geodesic curvature along each edge and the Gaussian curvature (zero) on each face
• Apply the Gauss-Bonnet formula to relate the curvature integrals to the Euler characteristic
  • Example: For a torus $T$, $\chi(T) = 0$, so $\int_T KdA = 0$
• Use the theorem to determine topological properties from geometric information, or vice versa
  • Example: If a closed surface has positive Gaussian curvature everywhere, then it must be topologically a sphere ($\chi = 2$)