In the taxi-cab plane show that ifA=(-5/2,2),B=(1/2,2), C=(2,2), P=(0,0), Q=(2,1) and R=(3,3/2)then A-B-C and P-Q-R. Show that line segment AB~to line segment PQ,line segment BC~line segmentQR, line segment AC~line segment PR. Sketch an appropriate picture.
Solution Summary
This shows how to sketch an appropriate picture for a given metric geometry situation.
Consider the infinite cylinder C with:
x = cos(theta)
y = sin(theta)
z = v
where
0 <= theta < 2 pi , - infinity < x < infinity.
a) Find the induced metric on C.
b) Find the geodesics on C.
Problem 1. Derive the formula given below for the Christoffel symbols ?_ij^k of a Levi-Civita connection in terms of partial derivatives of the associated metric tensor g_ij.
?_ij^k = (1/2) g^kl {?_i g_lj ? ?_l g_ij + ?_j g_il }.
Problem 2. Compute the Christoffel symbols of the Levi-Civita connection associated to ea
Task:
A. Discuss differences between neutral geometry and Euclidean geometry.
B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.
Note: Euclid's parallel postulate states the following: "For every line l and for every external
(See attached file for full problem description)
1. Show that the functions d defined below satisfy the properties of a metric.
a. Let X be any nonempty set and let d be defined by The d is the call the discrete metric.
b. If X is the set of all m-tuples of real numbers and, if for and , then (X,d) is a metric spac
Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?
A. Discuss differences between neutral geometry and Euclidean geometry.
B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.
