# Computing the Weingarten Map L by raising an index

• I
• Leo Mar
In summary, the conversation is discussing the proof that the Weingarten Map L for the unit sphere is equal to + or - the identity by computing the Lik in a coordinate patch and raising an index. The first and second fundamental forms have been computed and it has been found that L = g^-1Λ = (-1 0; 0 -1). The plus identity can be obtained by choosing a parametrization with an inward pointing normal. The concept of "raising an index" is not clearly defined in this context. The conversation also mentions the use of the metric tensor and the unit normal in calculating the Weingarten map.
Leo Mar
Hello,

I have to prove that the Weingarten Map L for the unit sphere is + or - the identity "by computing the Lik in a coordinate patch and raising an index".

S^2 : x(Φ,θ)=(sinΦcosθ, sinΦsinθ, cosΦ)

I have computed the first (g) and the second (Λ) fundamental forms and I have found :

L=g-1Λ= ( -1 0 ) = -Identity
_________( 0 -1 )

The plus identity is obtained similarly by choosing a parametrization with inward pointing normal.

But what does "raising an index" mean?

Thank you.

Well, we have basically no context to go off of, since you haven't told us what a Weingarten map is, or where you pulled this notation from.

But if the thing you call ##g## is the metric tensor (which I think is the same thing as the first fundamental form), then it certainly looks to me as though you have already "raised an index", which just means contracting a slot of a lower-index tensor with ##g^{-1}##.

Here is the general context. Although it seems to want a lowered not a raised index.

The second fundamental form is classically written as ##edx^{2} + 2fdxdy + gdy^{2}## in coordinates ##(x,y)##
The metric tensor is ##Edx^{2} + 2Fdxdy + Gdy^{2}##

You are being asked to write out the Wiengarten map in terms of these these two. If ##N## is the unit normal then for a tangent vector ##X## the Wiengarten map ##W(X)## is equal to ##X⋅N## , the derivative of ##N## with respect to ##X##.

Hint: Think of the second fundamental form as the tensor ##edx⊗dx +fdx⊗dy+fdy⊗dx+gdy⊗dy##
Think of the Weingarten map as a tensor.

Last edited:

## 1. What is the Weingarten Map L?

The Weingarten Map L is a mathematical concept used in differential geometry to describe the change in the normal vector of a surface as it moves along a given direction.

## 2. What does it mean to raise an index in the Weingarten Map L?

Raising an index in the Weingarten Map L refers to the process of transforming the normal vector of a surface from a covariant form to a contravariant form, or vice versa. This allows for easier calculation and manipulation of the vector.

## 3. How is the Weingarten Map L computed?

The Weingarten Map L can be computed by taking the partial derivatives of the surface's normal vector with respect to the surface's coordinates, and then using these derivatives to construct a matrix known as the shape operator. The shape operator is then used to calculate the Weingarten Map L for a given direction.

## 4. What is the significance of computing the Weingarten Map L?

The Weingarten Map L is an important tool in differential geometry as it allows for the calculation of curvature and other geometric properties of a surface. It is also used in physics to study the behavior of fluids and other materials on curved surfaces.

## 5. Are there any practical applications for computing the Weingarten Map L?

Yes, there are many practical applications for computing the Weingarten Map L. Some examples include analyzing the behavior of liquid crystals, understanding the shape and stability of soap bubbles, and studying the shape of biological membranes. It is also used in computer graphics and animation to create realistic 3D surfaces.

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