Ngô Quốc Anh

April 10, 2011

A property of the Weingarten map (the sharp operator)

Filed under: Riemannian geometry — Ngô Quốc Anh @ 23:29

Let M be a surface, p \in M and \eta the unit normal vector (field) of M defined along a neighborhood W of p.

Definition (the Weingarten map). The Weingarten map or the sharp operator L: T_pM \to T_pM is defined to be

\displaystyle L(X)=\nabla_X\eta

for any X\in T_pM.

To see this definition is well-defined, in this note, we shall prove that indeed L(X) \in T_pM for any tangent vector X\in T_pM. To this purpose, we have to show that

\langle \nabla_X\eta,\eta \rangle =0.

Be means of the normality, it holds \langle \eta,\eta \rangle =1 which implies

\nabla_X\langle\eta,\eta \rangle =0.

By the property saying that the connection \nabla is compatible with the metric, it holds

\nabla_X\langle\eta,\eta \rangle =\langle\nabla_X\eta,\eta \rangle +\langle\eta,\nabla_X\eta \rangle.


2\langle\nabla_X\eta,\eta \rangle=0.

The proof is complete.

The role of the Weingarten map is important since it is a tool to introduce the second fundamental form II:T_pM \times T_pM \to \mathbb R given by

II(X,Y)=\langle L(X),Y\rangle .

Apparently, for a plan, the second fundamental form is identically zero and for a sphere S of radius r, its second fundamental form is nothing but \frac{1}{r} {\rm id}.


  1. Notice that the Weingarten map is known as the shape operator. This has nothing to do with sharp!

    Comment by OrbiculaR — April 12, 2011 @ 4:27

    • Thank you, this is a typo :(.

      Comment by Ngô Quốc Anh — April 12, 2011 @ 4:29

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