9.3 Second fundamental form and the Chern-Lashof theorem

The second fundamental form is a key tool for understanding surface geometry. It measures how a surface bends in space, giving us info on curvature and shape. This form helps us calculate important values like Gaussian and mean curvature.

The Chern-Lashof theorem connects a surface's curvature to its topology. It shows that a surface's total absolute curvature is linked to its Betti numbers, which count the surface's "holes". This theorem is especially useful for studying convex shapes.

Second Fundamental Form and Geometry

Definition and Properties

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• The second fundamental form (denoted as II or h) is a quadratic form on the tangent space of a surface that measures how the surface bends in the ambient space
• For a hypersurface M in Euclidean space, the second fundamental form at a point p is defined as $II(X,Y) = -<dN(X),Y>$, where:
  • N is the unit normal vector field
  • X and Y are tangent vectors
  • $<,>$ denotes the inner product
• Geometrically, the second fundamental form describes the shape operator (or Weingarten map) of the surface, which maps tangent vectors to their normal components of the directional derivatives
• The eigenvalues of the second fundamental form are the principal curvatures, and the corresponding eigenvectors are the principal directions

Gaussian and Mean Curvature

• The determinant of the second fundamental form is the Gaussian curvature (K), which measures the intrinsic curvature of the surface
  • For example, a sphere has positive Gaussian curvature, a plane has zero Gaussian curvature, and a hyperbolic paraboloid has negative Gaussian curvature
• The trace of the second fundamental form (up to a factor of 1/2) is the mean curvature (H), which measures the extrinsic curvature of the surface
  • For example, a cylinder has zero mean curvature, while a sphere has positive mean curvature
• The Gaussian and mean curvatures are important invariants of the surface that characterize its local geometry and shape

Weingarten and Codazzi-Mainardi Equations

Weingarten Equations

• The Weingarten equations relate the shape operator (or Weingarten map) to the second fundamental form and the metric tensor of the surface
• For a hypersurface M with shape operator S and metric tensor g, the Weingarten equations state that $S(X) = -∇_X N$, where:
  • $∇$ is the Levi-Civita connection
  • N is the unit normal vector field
• The Weingarten equations allow us to express the shape operator in terms of the covariant derivative of the normal vector field
• They provide a way to compute the principal curvatures and principal directions of the surface using the second fundamental form and the metric tensor

Codazzi-Mainardi Equations

• The Codazzi-Mainardi equations are compatibility conditions between the second fundamental form and the Levi-Civita connection of the surface
• For a hypersurface M with second fundamental form h and Levi-Civita connection $∇$, the Codazzi-Mainardi equations state that $(∇_X h)(Y,Z) = (∇_Y h)(X,Z)$, where X, Y, and Z are tangent vectors
  • This equation can also be written as $(∇_X S)(Y) = (∇_Y S)(X)$, where S is the shape operator
• The Codazzi-Mainardi equations ensure that the second fundamental form is compatible with the metric and the connection of the surface
• They are necessary and sufficient conditions for a symmetric bilinear form to be the second fundamental form of a hypersurface
  • For example, given a symmetric bilinear form on a surface, one can check if it satisfies the Codazzi-Mainardi equations to determine if it can be realized as the second fundamental form of an embedded hypersurface

Chern-Lashof Theorem on Curvature

Statement and Definitions

• The Chern-Lashof theorem relates the total absolute curvature of a compact hypersurface to its topology, specifically the sum of its Betti numbers
• For a compact hypersurface M immersed in Euclidean space, the total absolute curvature $τ(M)$ is defined as the integral of the absolute value of the Gaussian curvature K over the surface: $τ(M) = ∫_M |K| dA$
• The Betti numbers $β_i(M)$ are topological invariants that count the number of independent i-dimensional holes in the surface
  • For example, a torus has Betti numbers $β_0 = 1$, $β_1 = 2$, and $β_2 = 1$, corresponding to one connected component, two independent loops, and one void

Theorem and Proof

• The Chern-Lashof theorem states that for a compact hypersurface M, the total absolute curvature $τ(M)$ is greater than or equal to $2∑_{i=0}^{n-1} β_i(M)$, where $β_i(M)$ are the Betti numbers of M
• The proof of the Chern-Lashof theorem involves the use of Morse theory and the Gauss-Bonnet theorem
  • Morse theory relates the critical points of a height function on the surface to its topology
  • The Gauss-Bonnet theorem relates the integral of the Gaussian curvature to the Euler characteristic of the surface
• The main idea of the proof is to construct a height function on the hypersurface and analyze its critical points using Morse theory
  • The absolute value of the Gaussian curvature at each critical point contributes to the total absolute curvature
  • The Morse inequalities relate the number of critical points to the Betti numbers of the surface
• Combining these results, one obtains the lower bound on the total absolute curvature in terms of the Betti numbers

Chern-Lashof Theorem for Hypersurfaces

Convex Hypersurfaces

• A hypersurface is called convex if it lies on one side of its tangent plane at every point
  • Examples of convex hypersurfaces include spheres, ellipsoids, and convex polyhedra
• For a compact convex hypersurface M, the Chern-Lashof theorem implies that the total absolute curvature $τ(M)$ is equal to $2∑_{i=0}^{n-1} β_i(M)$
  • This is because for a convex hypersurface, the Gaussian curvature is always non-negative, so the absolute value can be dropped
• For a compact convex hypersurface M, the Betti numbers are $β_0(M) = 1$ and $β_i(M) = 0$ for $i > 0$, as convex hypersurfaces are topologically equivalent to a ball
• Consequently, the Chern-Lashof theorem implies that for a compact convex hypersurface M, the total absolute curvature $τ(M)$ is equal to 2

Applications and Examples

• The Chern-Lashof theorem can be used to characterize compact convex hypersurfaces based on their total absolute curvature
  • For example, if a compact hypersurface has total absolute curvature equal to 2, it must be convex and topologically equivalent to a ball
• The theorem also provides a way to estimate the total absolute curvature of a hypersurface based on its topological properties
  • For example, if a compact hypersurface has non-trivial Betti numbers, its total absolute curvature must be greater than 2
• The Chern-Lashof theorem has applications in the study of the geometry and topology of submanifolds, as well as in the theory of integral geometry and geometric probability
  • For example, it can be used to derive bounds on the expected value of the total absolute curvature of random hypersurfaces