A torus of revolution with tube radius and centre-circle radius has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Curvature

Problem

A torus of revolution with tube radius

r

and centre-circle radius

R>r

has, in the angle

\theta

round the tube (

\theta=0

on the outer equator) and the angle

\phi

round the axis, the metric

\mathrm{d}s^2=r^2\,\mathrm{d}\theta^2+(R+r\cos\theta)^2\,\mathrm{d}\phi^2

. ((a)) Compute the Gaussian curvature

K(\theta)

from the Riemann tensor of this metric. ((b)) Say where on the torus

K

is positive, negative and zero, and evaluate

K

on the outer and inner equators of a ring doughnut with

R=3.0\ \mathrm{cm}

and

r=1.0\ \mathrm{cm}

. ((c)) Integrate

K

over the outer half (

|\theta|<\pi/2

), over the inner half, and over the whole torus. Compare the outer half's total with that of a whole sphere.

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