Question

Prove the following structure equations for M (the forms ω1, ω2, ωij , i, j = 1, . . . 3):

dω1 = ω12 ∧ ω2; dω2 = ω1 ∧ ω12;

dω12 = −ω13 ∧ ω23,

dω13 = ω12 ∧ ω23; dω23 = −ω12 ∧ ω13.

.3 the folowing structure nicture equations for M (the forms-i-2 -u i , were defined in the notes): ia13

Answer 1

Proof for given structure equations for M: For M_1 the field of orthonormal frames are x, e_1, e_2 and e_3 such that t belongs to M(x elementof M) Where e_3 is the unit normal at x. e_1 and e_2 are along principal directions. Then, fundamental equations for M have the following: dt = w_1 e_1 + w_2 e_2 ---(1) de_1 = w_12 e_2 + w_13 e_3 ---(2) de_2 = -w_12 e_1 + w_23 e_3 ---(3) de_3 = -w_13 e_1 - w_23 e_2 ---(4) At the choice of frame and cartan's lemma allows for the two principal curvatures a and c they can be written as following w_12 = hw_1 + kw_2 ---(5) w_13 = aw_1 ---(6) w_23 = cw_2 ---(7) If a > c in h = 1/2 (a + c) ---(8) Where h is mean curvature of M k is the Gaussian curvature k = a * c ---(9) Hence, the forms show in equation (1), (2), (3), (4) satisfies the fundamental structure equations which are