Case of Equality
The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called equilateral torus, i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in .
Read more about this topic: Loewner's Torus Inequality
Famous quotes containing the words case and/or equality:
“The circumstances with which every thing in this world is begirt, give every thing in this world its size and shape;—and by tightening it, or relaxing it, this way or that, make the thing to be, what it is—great—little—good—bad—indifferent or not indifferent, just as the case happens.”
—Laurence Sterne (1713–1768)
“Let the same mind be in you that was in Christ Jesus, who, though he was in the form of God, did not regard equality with God as something to be exploited, but emptied himself, taking the form of a slave, being born in human likeness. And being found in human form, he humbled himself and became obedient to the point of death M even death on a cross.”
—Bible: New Testament, Philippians 2:5-8.