In mathematics, a LeBrun manifold is a
connected sum of copies of the
complex projective plane, equipped with an explicit self-dual metric. Here, self-dual means that the
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
is its own
Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the al ...
. The metric
is determined by the choice of a finite collection of points in
hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
. These metrics were discovered by , and named after LeBrun by .
References
*
*{{Citation , last1=LeBrun , first1=Claude , authorlink=Claude LeBrun , title=Explicit self-dual metrics on CP
2#...#CP
2 , url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214446999 , mr=1114461 , year=1991 , journal=Journal of Differential Geometry , issn=0022-040X , volume=34 , issue=1 , pages=223–253
Differential geometry