本科高年级Geometry & Topology计划中

Differential geometry of curves and surfaces

Curvature and torsion of space curves, the first and second fundamental forms of surfaces, Gaussian and mean curvature, Gauss's Theorema Egregium, geodesics and the Gauss–Bonnet theorem.

math.geom.curves-surfaces

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前置知识

Multivariable and vector calculus

37 项必需的前置知识（一直追溯到底）

你将学会

Compute curvature and torsion using the Frenet–Serret frame.
Compute the first and second fundamental forms and the Gaussian curvature of standard surfaces.
State the Theorema Egregium and the Gauss–Bonnet theorem and check them on spheres and tori.

定义

The curvature of a unit-speed curve γ is κ = |γ″|.
The Gaussian curvature K of a surface at a point is the product of its principal curvatures.
A geodesic on a surface is a curve whose acceleration is everywhere normal to the surface.

定理

Theorema Egregium: Gaussian curvature depends only on the first fundamental form, so it is intrinsic.
Gauss–Bonnet: for a compact oriented surface S without boundary, ∫_S K dA = 2πχ(S).
No map from an open region of a sphere to the plane preserves all distances, because their Gaussian curvatures differ.

常见误解

Thinking a cylinder is curved in the Gaussian sense; its Gaussian curvature is zero.
Confusing the curvature of a curve with the Gaussian curvature of a surface.

可选发现

Smooth manifolds The Theorema Egregium motivates intrinsic geometry, which is studied on abstract manifolds with no surrounding space.
← Euclidean geometry and proof

学习路径

Undergraduate mathematics major