Consider Euclidean R2 with the standard cartesian coordinates x1, x2 and metric ds? = (dx?)2 + (dx2). Since the metric is constant in these coordinates, the connection vanishes Tuv = 0, and the curvature vanishes, Rouy = 0.

Now change to polar coordinates r, o

a.Using the transformation rule we derived in class for the connection,

convert the cartesian connection into polar coordinates.

c. Check your answer to part a. by computing the connection directly from the metric in polar coordinates,

ds2 = dr2 + r2a02,

(8) using the formula for the connection in terms of the metric:

c. Compute the Riemann curvature in polar coordinates directly from the connection using the expression (3) and verify that it vanishes (as it must since this is just flat space).