Let be a surface, and the unit normal vector (field) of defined along a neighborhood of .
Definition (the Weingarten map). The Weingarten map or the sharp operator is defined to be
for any .
To see this definition is well-defined, in this note, we shall prove that indeed for any tangent vector . To this purpose, we have to show that
Be means of the normality, it holds which implies
By the property saying that the connection is compatible with the metric, it holds
The proof is complete.
The role of the Weingarten map is important since it is a tool to introduce the second fundamental form given by
Apparently, for a plan, the second fundamental form is identically zero and for a sphere of radius , its second fundamental form is nothing but .