# Derivative of surface normal || tangent unit vector

Let be w=w(σ) a curve on a surface parametrized by the arc length (the natural parametrization). Consider the m surface normal along this curve as the function of the σ arc length of the curve. Prove that m'(σ) is parallel to the t(σ) tangent unit vector of the curve for all σ, IFF this curve is a curvature line.

I would appreciate if you could help me!

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#### lavinia

Gold Member
Let be w=w(σ) a curve on a surface parametrized by the arc length (the natural parametrization). Consider the m surface normal along this curve as the function of the σ arc length of the curve. Prove that m'(σ) is parallel to the t(σ) tangent unit vector of the curve for all σ, IFF this curve is a curvature line.

I would appreciate if you could help me!
What definition of curvature line do you use?

What definition of curvature line do you use?
In my terminology curvature line is a curve on a surface whose tangents are always in the direction of principal curvature.

#### lavinia

Gold Member
In my terminology curvature line is a curve on a surface whose tangents are always in the direction of principal curvature.

To me your equation is the definition of a line of curvature, that is a curve that points in the direction of an eigen vector of the second fundamental form.

To me your equation is the definition of a line of curvature, that is a curve that points in the direction of an eigen vector of the second fundamental form.
To tell the truth, I don't really see the point. Curvature line is a synonym of line of curvature.
The direction of an eigen vector of the second fundamental form is the same as the direction of the principal curvature.
How does the statement follow from the definition?

#### lavinia

Gold Member
To tell the truth, I don't really see the point. Curvature line is a synonym of line of curvature.
The direction of an eigen vector of the second fundamental form is the same as the direction of the principal curvature.
How does the statement follow from the definition?
An eigen vector by definition means X.N = kX where N here is the unit normal.

On the other hand if you define a direction of principal curvature to be the direction of a curve of extremal curvature of all curves that lie on the intersection of the surface with a plane that contains the unit normal, then you need to show that such a direction is an eigen vector and visa versa. This must be what your question really is.

Since the unit normal,N, is orthogonal to X, <N,X> = 0 for any tangent direction X.

So 0 = X.<N,X> = <X.N,X> + <N,X.X> where X. means the derivative along a curve fitting X.

If X has length 1, then the second term is the curvature of a curve fitting X. So an extremal of the first term must be an extremal of the second.

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Thank you very much for your detailed answer. I am afraid I don't really understand your notation. Does X. denote derivative, dot products or just a fullstop?
Thanks once more!

#### lavinia

Gold Member
Thank you very much for your detailed answer. I am afraid I don't really understand your notation. Does X. denote derivative, dot products or just a fullstop?
Thanks once more!
On a surface, a function can be differentiated along curves. One just looks at the function as it varies with respect to the curve's parameter as you did using the parameter of arc length.

If X is a tangent vector at a point on the surface then X.f denotes the derivative of f
at that point along a curve whose velocity vector is X. Since vector fields, such as the unit normal to the surface, are just triples of functions, they can also be differentiated along a curve on the surface. So X.N means the derivative of the unit normal vector field with respect to a curve whose velocity is X. If X has length one, then this curve is parameterized by arc length as in your original formulation.

The derivative of the function that is the inner product (dot product) of two vector fields satisfies the Leibniz formula

X.<V,W> = <X.V,W> + <V,X.W>. ( <,> just means dot product.)

Note that the left hand side is the derivative of a function while the right hand side is the sum of inner products of derivatives of vector fields with vector fields.

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