Extremal Length of Topologically Essential Paths in Projective Plane
In the above examples, the extremal which maximized the ratio and gave the extremal length corresponded to a flat metric. In other words, when the Euclidean Riemannian metric of the corresponding planar domain is scaled by, the resulting metric is flat. In the case of the rectangle, this was just the original metric, but for the annulus, the extremal metric identified is the metric of a cylinder. We now discuss an example where an extremal metric is not flat. The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map . Let denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in is obtained by projecting a curve on the sphere from a point to its antipode.) Then the spherical metric is extremal for this curve family. (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is .
Read more about this topic: Extremal Length, Examples
Famous quotes containing the words plane, paths, length and/or essential:
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)
“Now the hungry lion roars,
And the wolf behowls the moon;
Whilst the heavy ploughman snores,
All with weary task fordone.
Now the wasted brands do glow,
Whilst the screech-owl, screeching loud,
Puts the wretch that lies in woe
In remembrance of a shroud.
Now it is the time of night,
That the graves, all gaping wide,
Every one lets forth his sprite,
In the church-way paths to glide:”
—William Shakespeare (1564–1616)
“Rivers must have been the guides which conducted the footsteps of the first travelers. They are the constant lure, when they flow by our doors, to distant enterprise and adventure; and, by a natural impulse, the dwellers on their banks will at length accompany their currents to the lowlands of the globe, or explore at their invitation the interior of continents.”
—Henry David Thoreau (1817–1862)
“Today, each artist must undertake to invent himself, a lifelong act of creation that constitutes the essential content of the artist’s work. The meaning of art in our time flows from this function of self-creation. Art is the laboratory for making new men.”
—Harold Rosenberg (1906–1978)