Only part B, C and D.

Differential Forms / Covariant Derivative / Riemannian metric

2. Consider the following Riemannian metric on R2: g=(1+ y)d.tdx + xy(dx dy + dy de) + (1+x2)dy dy (1) a. If w1 and wa are l-forms defined by w = V1 + y2dr + zdy, +y w? 1 + x2 + y2 -dy | + y2 then show that the Riemannian metric (1) takes the form g=wiwt+w24 b. Find a 1-form such that dwl dw2 -Ow2 \ @ c. Find the Gaussian curvature. d. Let X= 1 a V1+ year Y = 1+ y2 1 + 2 + y2 a 1 + y2 x a . Show that VXY - VyX = (X,Y] 2. Consider the following Riemannian metric on R2: g=(1+ y)d.tdx + xy(dx dy + dy de) + (1+x2)dy dy (1) a. If w1 and wa are l-forms defined by w = V1 + y2dr + zdy, +y w? 1 + x2 + y2 -dy | + y2 then show that the Riemannian metric (1) takes the form g=wiwt+w24 b. Find a 1-form such that dwl dw2 -Ow2 \ @ c. Find the Gaussian curvature. d. Let X= 1 a V1+ year Y = 1+ y2 1 + 2 + y2 a 1 + y2 x a . Show that VXY - VyX = (X,Y]