Definition
Let be a compact smooth manifold of dimension . The normalized Einstein–Hilbert functional assigns to each Riemannian metric on a real number as follows:
where is the scalar curvature of and is the volume form associated to the metric . Note that the exponent in the denominator is chosen so that the functional is scale-invariant. We may think of as measuring the average scalar curvature of over . It was conjectured by Yamabe that every conformal class of metrics contains a metric of constant scalar curvature (the so-called Yamabe problem); it was proven by Yamabe, Trudinger, Aubin, and Schoen that a minimum value of is attained in each conformal class of metrics, and in particular this minimum is achieved by a metric of constant scalar curvature. We may thus define
where the infimum is taken over the smooth functions on . The number is sometimes called the conformal Yamabe energy of (and is constant on conformal classes).
A comparison argument due to Aubin shows that for any metric, is bounded above by, where is the standard metric on the -sphere . The number is equal to and is often denoted . It follows that if we define
where the supremum is taken over all metrics on, then (and is in particular finite). The real number is called the Yamabe invariant of .
Read more about this topic: Yamabe Invariant
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)