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## 5.9 The Principal Curvatures of a Surface

Let us discuss in more detail the properties of a surface connected with its second fundamental form. We now denote the coordinates without indices, using u=u 1, v=u 2. We also use subscripts u, v to indicate partial differentiation:

Hence the second fundamental form is

where

Consider the first differential of n at a point P:

Since n is a unit vector its differential d n is orthogonal to n and thus lies in the osculating plane to the surface at P. We know that the differential d r= r u du+ r v dv also lies in the plane osculating at P. Let us consider the relation between the differentials d r and d n with respect to the variables du, dv now considered as independent variables. It is clearly a linear correspondence d r d n. Thus it defines a tensor A in two-dimensional space such that

This tensor is completely defined by its values n u= A r u and n v= A r v.

#### Lemma 5.1

The tensor A is symmetric.

Proof. It is enough to establish the equality

for a pair of linearly independent vectors ( x 1, x 2). To show symmetry of A consider

Similarly

The symmetry of A follows from the identity

this is derived by...

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