Conormal

Conormal refers to a construction in differential and algebraic geometry that captures covectors orthogonal to a submanifold. Given a smooth manifold X and a smooth submanifold Y, the conormal bundle N*_{Y/X} is the subset of the restricted cotangent bundle T*X|_Y consisting of covectors that vanish on the tangent space TY. The fiber at y in Y is N*_y Y = {ξ in T*_y X : ξ(v) = 0 for all v in T_y Y}. Equivalently, N*_{Y/X} is the annihilator of TY inside T*X|_Y. The conormal bundle is a Lagrangian submanifold of T*X with its canonical symplectic form and is dual to the normal bundle N_Y = TX|_Y/TY.

If Y is locally defined by equations, the conormal is generated by the differentials of these defining functions. For a hypersurface defined by f = 0, N*_{Y/X} is locally spanned by df, so covectors proportional to df_y form the conormal fiber at y.

In microlocal analysis and the theory of distributions, the conormal bundle describes singularities along Y. A distribution is conormal to Y if its singularities are controlled by covectors in N*_{Y/X}, and conormal distributions have wavefront sets contained in N*_{Y/X}. This framework is central to understanding propagation of singularities for partial differential equations, especially in boundary value problems where the geometry of Y influences how singularities move.

In algebraic geometry, the conormal sheaf N^*_{Y/X} is the annihilator of TY inside the restricted sheaf of differentials Ω^1_X|_Y, and the conormal variety is the spectrum of the symmetric algebra of N^*_{Y/X}. For smooth Y, N^*_{Y/X} is dual to the normal bundle, expressing the infinitesimal failure of Y to be tangent to X.