Riemannian Geometry on Quantum Spaces

Riemannian Geometry on Quantum Spaces

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended ∗email address: [email protected] to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.

Riemannian Geometry on Loop Spaces

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter. In Part I, we compute the Levi-Civita connection for these metrics. The connection and curvature forms take values in pseudodifferential operators (ΨDOs), and we compute the top symbols of these forms. In Part II, we develop a theory of Chern-Simons classes CS ...

Riemannian Geometry of Quantum Computation

An introduction is given to some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group SU(2n). Using the Lie algebra su(2n), detailed derivations are given of a useful Riemannian geometry of SU(2n), including the connection, curvature, the geodesic equation for minimal complexity quantum compu...

Riemannian Geometry of Quantum Groups

On the Riemannian Geometry of Seiberg-witten Moduli Spaces

We construct a natural L-metric on the perturbed Seiberg-Witten moduli spaces Mμ+ of a compact 4-manifold M , and we study the resulting Riemannian geometry of Mμ+ . We derive a formula which expresses the sectional curvature of Mμ+ in terms of the Green operators of the deformation complex of the Seiberg-Witten equations. In case M is simply connected, we construct a Riemannian metric on the S...