And the process continues into successively higher dimensions.
Now comes the beautiful connection with Lie theory.
At every point along the curve, the n perpendicular frame vectors form an orthonormal frame. As the curve evolves, this entire frame rotates continuously.
Those rotations belong to the Lie group SO(n).
If we collect the frame vectors into a matrix F(s), its evolution can be written:
F′(s) = F(s) Ω(s)
Here Ω(s) is a skew-symmetric matrix:
Ωᵀ = −Ω
And skew-symmetric matrices are precisely the elements of the Lie algebra so(n).
The remarkable part is that the generalised curvatures
κ₁, κ₂, …, κₙ₋₁
appear directly inside Ω.
So we arrive at a beautiful change of viewpoint:
A geometrical curve can be understood through the evolution of a rotating frame in a Lie group, while its curvatures appear as the coefficients governing that motion in the corresponding Lie algebra.
In 3D, those quantities are curvature and torsion.
In higher dimensions, they become a hierarchy of generalised curvatures describing how a curve progressively explores additional dimensions.
Geometry, differential equations and Lie theory are different languages for describing the same motion.
#DifferentialGeometry #LieGroups #LieAlgebra #FrenetSerret #Curvature #Torsion #Geometry #Mathematics #HigherDimensions
