### Differential geometry notes 4

#### by exactnature

* Formulae of Frenet:* These describe the derivatives of the moving trihedron of a curve in terms of the moving trihedron itself:

Note that the coefficients inside the skew-symmetric matrix need not be constant, hence the system described by this equation is not necessarily linear.

* Definition 16.1:* A vector is called a

**rotation vector**of a rotation if it has the following properties:

- has the direction of the axis of rotation.
- The sense of is such that the rotation has the clockwise sense if one looks from the initial point of to its terminal point.
- The magnitude of equals the angular velocity of the rotation, that is, the velocity of points at distance 1 from the axis of rotation.

* Theorem 16.2:* The rotation vector of the trihedron of a curve of class with non-vanishing curvature, when a point moves along with constant velocity 1 is given by the expression:

This vector is called **the vector of Darboux**.

The curves traced out by the terminal points of , and on the unit sphere as the trihedron of a curve moves are called **the tangent**, **binormal** and **principal normal indicatrix** respectively. The linear elements of these curves , and are given by:

Hence, we can write (Equation of Lancret).

* Shape of a curve in the neighborhood of any of its points:* We can Taylor expand the representation of the curve around (note that the zero point can be chosen arbitrarily):

We have , , and . We can choose the coordinate system such that , , .

Plugging these expressions in the Taylor expansion, we get:

where and are the values of the curvature and torsion at . If we keep only the leading terms, we get:

(, )