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the Frenet-Serret Formulas |
In differential geometry there is a beautiful result that was discovered in the middle of the 19th century independently by Jean Fréderic Frenet and Joseph Alfred Serret. They specified for any smooth curve a set of three orthogonal unit vectors: 1. T, the unit tangent vector to the curve, 2. N, the unit normal vector to the curve, 3. B, the unit binormal vector defined as T×N. The rates of change of these vectors with respect to arc length s then takes the simple form
where κ is the curvature and τ is the torsion of the curve.
About 1880 Jean Gaston Darboux generalized this result to surfaces and around 1900 Élie Cartan fully generalized it.
A frame at a point p is a triplet of orthogonal unit length vectors, say E_{1}, E_{2}, and E_{3} such that E_{i}·E_{j}=δ_{ij} for all i and j from 1 to 3. A frame field is simply frames defined at every point of Euclidean 3-space.
There is a natural frame field for Euclidean 3-space, the unit vectors in the x, y and z directions.
A frame is a basis, therefore any vector W at p can be represented as a linear sum of the vectors of the frame; i.e., there exist real valued coefficients {c_{1},c_{1},c_{3}} such that
Consider a scalar field ψ(p) defined for all points p of Euclidean 3-space. It can also be expressed as ψ(x(p),y(p),z(p)). The scalar is given along a line p+tv as ψ(t) and the rate of change in the direction v is given as
The terms on the RHS of the above can be expressed as (∇ψ)·v. Thus the derivative of ψ(p) in the direction v, ∇_{v}(ψ), is ∇ψ·v.
Note that if a scalar field ψ(p) happens to be the dot-product of two vectors fields, say a(p) and b(p), then
Let W(p) be a vector field. The covariant derivative of W(p) with respect to a vector v is the vector of the rates of change of W(p) as the point p starts moving in the direction v. This is expressed as
For a specified value of v, ∇_{v}(W(p)) is a vector field. Then necessarily there exist a field of coefficients {c_{1}(p),c_{2}(p),c_{3}(p)} such that
∇_{v}(W(p)) = Σc_{i}(p)E_{i}(p)
where the summation
is over i from 1 to 3
Because of the orthonormality (orthogonal vectors of unit length) of the frame field the coefficients are given by
The covariant derivatives can be determined for each of the unit vectors E_{1}(p), E_{2}(p) and E_{3}(p). There will be three sets of coefficients each having three components; i.e.,
Note that ω_{ij}=∇_{v}E_{i}(p)·E_{j}(p) for all i and j and all v and all p.
So far nothing profound has been established. The important matter is the relationships among the coefficients ω_{ij}.
Note now that ∇_{v}(δ_{ij})=0 for all i and j and all v and p. But E_{i}(p)·E_{j}(p)=δ_{ij}. Therefore
Since ω_{ij}(p,v)=∇_{v}E_{i}(p)·E_{j}(p) the previous equation reduces to:
For i=j this last expression implies ω_{ii}(p,v)=0 for all i.
Thus the matrix of the coefficients ω_{ij}(p,v) is skew-symmentric.
This is the essential result of Élie Cartan.
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