18.950 Differential Geometry, Lecture 6 Notes | Lecture Note - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.950 | Differential Geometry
Academic year:

2020
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371

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1
Author:

Hidalgokr1

Lecture 6 Take a Frenet curve c in Rn . Let E(t) be the matrix with columns e1 (t), . . . , en (t). Theorem 6.1. We have Here κ1 (t), . . . , κn−2 (t) > 0, and κn−1 (t) ∈ R. Concretely, The functions κi (t) are called the Frenet curvatures of c. Physically, they are again of type 1/m. As usual they are reparametrization invariant. Proposition 6.2. Let c be a Frenet curve in Rn . Then n−1 = κni−i . i=1 Example 6.3. A regular plane curve is always Frenet. The Frenet basis is is the ordinary curva ture, and the Frenet equations of motion reduce to Proposition 1.3. Example 6.4. Let c : I → R3 be a space curve, parametrized with unit speed. This is Frenet if and only if The Frenet basis is κ = κ1 is called the curvature and τ = κ2 the torsion. Concretely The Frenet equations are

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