G

Gδ set A countable intersection of open sets. See Fσ set.

Gauss equations A system of partial differential equations arising in the theory of surfaces. If M is a surface in R3 with local coordinates (u1, u2), its geometric invariants can be described by its first fundamental form gij (u

1, u2), and second fundamental form Lij (u

1, u2). The Christoffel symbols kij are determined by the first fundamental form. In order for functions gij and Lij , i, j = 1, 2 to be the first and second fundamental forms of a surface, certain integrability conditions (arising from equality of mixed partial derivatives) must be satisfied. One set of conditions, the Gauss equations, relate the determinant of the second fundamental form to an expression involving only the first fundamental form (and its first and second partial derivatives). See also Christoffel symbols, first fundamental form.