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Élie Cartan's Generalization of
the Frenet-Serret Formulas

The Frenet Serret Formulas for a Curve

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

| dT/ds |    | 0 κ 0 | | T |
| dN/ds | = | -κ 0 τ | | N |
| dB/ds |    | 0 -τ 0 | | B |

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.

Frame Fields

A frame at a point p is a triplet of orthogonal unit length vectors, say E1, E2, and E3 such that Ei·Ejij 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 {c1,c1,c3} such that

W(p) = Σi=13ciEi(p).

The Directional Derivatives of a Scalar Field

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

dψ/dt = ∇v(ψ)
= (∂ψ/∂x)(dx/dt) + (∂ψ/∂y)(dy/dt) + (∂ψ/∂z)(dz/dt)

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

v(a(p)·b(p)) = ∇v(a(p))·b(p)) + a(p)·∇v(b(p)

The Covariant Derivative of a Vector Field

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

v(W(p)) = d/dt(W(p)+tv) for t=0

For a specified value of v, ∇v(W(p)) is a vector field. Then necessarily there exist a field of coefficients {c1(p),c2(p),c3(p)} such that

v(W(p)) = Σci(p)Ei(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

ci(p) = ∇v(W(p))·Ei(p)

The Connections for the Frame Field

The covariant derivatives can be determined for each of the unit vectors E1(p), E2(p) and E3(p). There will be three sets of coefficients each having three components; i.e.,

| ∇vE1(p) |      | ω1,1(p,v) ω1,2(p,v) ω1,3(p,v) | | E1 |
| ∇vE1(p) | = | ω2,1(p,v) ω2,2(p,v) ω2,3(p,v) | | E2 |
| ∇vE1(p) |      | ω3,1(p,v) ω3,2(p,v) ω3,3(p,v) | | E3 |

Note that ωij=∇vEi(p)·Ej(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 ωij.

Note now that ∇vij)=0 for all i and j and all v and p. But Ei(p)·Ej(p)=δij. Therefore

v(Ei(p)·Ej(p)) = ∇vEi(p)·Ej(p) + Ei(p)·∇vEj(p) = 0
or, upon reversing the order
of terms in the last expression
v(Ei(p)·Ej(p)) = ∇vEi(p)·Ej(p) + ∇vEj(p)·Ei(p) = 0

Since ωij(p,v)=∇vEi(p)·Ej(p) the previous equation reduces to:

ωij(p,v) + ωji(p,v) = 0
or, equivalently
ωji(p,v) = −ωij(p,v)

For i=j this last expression implies ωii(p,v)=0 for all i.

Thus the matrix of the coefficients ωij(p,v) is skew-symmentric.

| ∇vE1(p) |     | 0                  ω1,2(p,v)     ω1,3(p,v) | | E1(p) |
| ∇vE1(p) | = | −ω1,2(p,v)        0             ω2,3(p,v) | | E2(p) |
| ∇vE1(p) |      | −ω1,3(p,v)      −ω2,3(p,v)             0 | | E3(p) |

This is the essential result of Élie Cartan.

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