APM3713 Assignment 6 2025 | 100% COMPLETE SOLUTIONS

APM3713
ASSIGNMENT 6 2025

UNIQUE NO.
DUE DATE: SEPTEMBER 2025

, lOMoARcPSD|21997160




APM3713 Assignment 6 2025

Chapters 3 and 4

Question 1

Consider the surface that can be parameterized as
√
x (u, v) = u2 + 1 cos v
√
y (u, v) = u2 + 1 sin v
z (u, v) =u


(a) Let x 1 = u and x 2
= v. Find the line element for the surface.

(b) What is the metric tensor and the dual metric tensor?

(c) Determine the values of all the Christoffel coefficients of the surface.

(d) What is the value of the component R 1212 of the Riemann curvature tensor?

(e) What is the Ricci tensor for the surface? Hint: For a 2 dimensional space with a diagonal metric
tensor, we have for the Riemann curvature tensor:
g11 g11
R 1212 = −R 1
221 = g22 R 2121 = − g22 R 2112


(f) What is the curvature scalar R for the surface?

(g) What is the Gaussian curvature of the surface?

(h) Is the surface Euclidean? Explain your answer.



Question 2

Show that if the metric g ij is diagonal, then Γ ikl = 0 whenever i, k and l are distinct, i.e. whenever
i ̸ = k ̸= l.



Question 3

Two N -dimensional Riemann spaces M andM̄ have the metric tensors g ij and ḡij respectively, and

ḡij = kgij

where k is a constant.What are the relationships between the curvature tensors, Ricci tensors, curvature
scalar and Einstein tensors of the two spaces?




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