Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

The index bundle for Fredholm morphisms

Abstract: We extend the index bundle construction for families of bounded Fredholm operators to morphisms between Banach bundles.

Fibre bundle formulation of nonrelativistic quantum mechanics. III. Pictures and integrals of motion

Abstract: We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. It's evolution is determined through the bundle (analogue of the) Schrodinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In this third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, there are found the state sections and bundle morphisms corresponding to state vectors and observables respectively. The equations of motion for these quantities are derived too. Using the results obtained, we consider from the bundle view-point problems concerning the integrals of motion. An invariant (bundle) necessary and sufficient conditions for a dynamical variable to be an integral of motion are found.

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Abstract: We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.

Finite rank vector bundles on inductive limits of grassmannians

Abstract: If P 1 is the projective ind-space, i.e. P 1 is the inductive limit of linear embeddings of complex projective spaces, the Barth-Van de Ven-Tyurin (BVT) Theorem claims that every finite rank vector bundle on P 1 is isomorphic to a direct sum of line bundles. We extend this theorem to general sequences of morphisms between projective spaces by proving that, if there are infinitely many morphisms of degree higher than one, every vector bundle of finite rank on the inductive limit is trivial. We then establish a relative version of these results, and apply it to the study of vector bundles on inductive limits of grassmannians. In particular we show that the BVT Theorem extends to the ind-grassmannian of subspaces commensurable with a fixed infinite dimensional and infinite codimensional subspace in C 1 . We also show that, for a class of twisted ind-grassmannians, every finite rank vector bundle is trivial. 2000 AMS Subject Classification: Primary 32L05, 14J60, Secondary 14M15.