A Differential Approach to Geometry by Francis Borceux

Author:Francis Borceux
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

This system can be further re-written as

For a fixed value (α 0,β 0) corresponding to a local maximum or minimum of κ n , let us then consider the following system in α, β.

This is a homogeneous system of two linear equations with two unknowns and we know that it admits a non-zero solution, namely, (α 0,β 0). This is possible if and only if the determinant of the system is zero (see any algebra course). Thus

This can be re-written as

This proves in particular that every locally maximal or locally minimal value of κ n (α,β) is a root of the equation

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