Schr\"odinger Manifolds

Understand

This article propounds, in the wake of influential work of Fefferman and Graham about Poincar\'e extensions of conformal structures, a definition of a (Poincar\'e-)Schr\"odinger manifold whose boundary is endowed with a conformal Bargmann structure above a non-relativistic Newton-Cartan spacetime.

• Examples of such manifolds are worked out in terms of homogeneous spaces of the Schr\"odinger group in any spatial dimension, and their global topology is carefully analyzed.
• These archetypes of Schr\"odinger manifolds carry a Lorentz structure together with a preferred null Killing vector field; they are shown to admit the Schr\"odinger group as their maximal group of isometries.
• The relationship to similar objects arising in the non-relativisitc AdS/CFT correspondence is discussed and clarified.