nLab conormal bundle

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 $S$ of an abelian category $A$ 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 $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ where $N\in Ob(S)$. One can show that if $T\subset S$ is an inclusion of topologizing subcategories, then $\mathcal{I}_{S}\subset \mathcal{I}_{T}$. In particular, for Gabriel multiplication of topologizing subcategories we have $\mathcal{I}_{S\circ S} \subset \mathcal{I}_S$.

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

Related concepts

• normal bundle

• conormal bundle