📂Geometry

Bingarten Map

Definition¹

Let $M$ be a surface, and $p \in M$ be a point on the surface. The map $L : T_{p}M \to \mathbb{R}^{3}$, defined as follows, is called the Weingarten map.

$$ L (\mathbf{X}) = - \mathbf{X}\mathbf{n} $$

Here, $\mathbf{X} \in T_{p}M$ is a tangent vector, $\mathbf{n}$ is a unit normal, and $\mathbf{X}\mathbf{n}$ is the directional derivative of $\mathbf{n}$.

Properties

1. $L$ is a linear transformation that is $L : T_{p}M \to T_{p}M$.

2. Since $\left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\}$ is a basis for $T_{p}M$, if we denote it as $L(\mathbf{x}_{k}) = \sum\limits_{l}{L^{l}}_{k}\mathbf{x}_{l}$, the following holds:

   $${L^{l}}_{k} = \sum_{i}L_{ik}g^{il} = \sum_{i}L_{ki}g^{il}$$

   Where, $L_{ij}$ is the coefficient of the second fundamental form, and $[g^{kl}]$ is the inverse matrix of the first fundamental form coefficients. When expressed as a matrix,

   $$ \begin{bmatrix} {L^{l}}_{k} \end{bmatrix} = \begin{bmatrix} {L^{1}}_{1} & {L^{1}}_{2} \\ {L^{2}}_{1} & {L^{2}}_{2} \end{bmatrix} = \begin{bmatrix} g^{li} \end{bmatrix} \begin{bmatrix} L_{ik} \end{bmatrix} $$

Explanation

The minus sign in the definition is there for convenience.

The Weingarten map can be understood as an operator that measures the rate of change of $\mathbf{n}$ in each tangent direction at each point $p$. For this reason, it is also referred to as a shape operator.

1. By definition, $L$ is defined as a map that sends $T_{p}M$ to $\mathbb{R}^{3}$, but in fact, it can be seen that it sends it to $T_{p}M$.

2. In other words, ${L^{l}}_{k}$ is the coefficient of the $l$-th basis of $L(\mathbf{x_{k}})$. That is, if expressed as a coordinate vector for the basis $B = \left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\}$, it is as follows. $$ L(\mathbf{x}_{k}) = {L^{1}}_{k}\mathbf{x}_{1} + {L^{2}}_{k}\mathbf{x}_{2} $$ $$ \left[ L(\mathbf{x}_{k}) \right]_{B} = \begin{bmatrix} {L^{1}}_{k} \\ {L^{2}}_{k} \end{bmatrix} $$ Therefore, the matrix representation of $L$ is as follows. $$ [L]_{B} = \begin{bmatrix} {L^{1}}_{1} & {L^{1}}_{2} \\ {L^{2}}_{1} & {L^{2}}_{2} \end{bmatrix} $$ Furthermore, due to the properties of the first fundamental form, the following holds. $$ L_{ij} = \sum_{l}L_{il}\delta_{lj} = \sum\limits_{l,k} L_{il}g^{lk}g_{kj} = \sum\limits_{l,k} L_{li}g^{lk}g_{kj} = \sum\limits_{k}{L^{k}}_{i}g_{kj} $$

Since $L$ is a linear transformation between finite-dimensional vector spaces, $\tr{L}$ and $\det(L)$ are invariants, and are respectively called the mean curvature and the Gaussian curvature.