nLab
conormal bundle

Contents

Definition

Of submanifolds

In differential geometry a conormal bundle of an embedded submanifold is the (fiberwise linear) dual of the normal bundle.

Of locally ringed subspaces

The phrase conormal bundle is also used for more general conormal sheaf in the study of locally ringed spaces, especially of analytic spaces and algebraic schemes.

Of abelian subcategories

Even more generally, Alexander Rosenberg defines a conormal bundle of a topologizing subcategory SS of an abelian category AA as follows.

He first modifies the notion of the defining sheaf of ideals of a closed subscheme to the notion of defining ideal of a topologizing subcategory as the endofunctor = SEnd(A)\mathcal{I}=\mathcal{I}_S\in End(A) which is the subfunctor of identity Id AId_A assigning to any MAM\in A the intersection of kernels Ker(f)Ker(f) of all morphisms f:MNf: M\to N where NOb(S)N\in Ob(S). One can show that if TST\subset S is an inclusion of topologizing subcategories, then S T\mathcal{I}_{S}\subset \mathcal{I}_{T}. In particular, for Gabriel multiplication of topologizing subcategories we have SS S\mathcal{I}_{S\circ S} \subset \mathcal{I}_S.

Then the conormal bundle is simply Ω S= S/ SS\Omega_S = \mathcal{I}_S/\mathcal{I}_{S\circ S}, similarly to the sheaf case.

Revised on May 12, 2011 17:00:07 by Urs Schreiber (131.211.238.109)