Isometric Spectral Subtriples

TL;DR

This paper explores the concept of spectral subtriples in noncommutative geometry, proposing a new definition based on submanifold algebra and demonstrating its applicability in certain examples.

Contribution

It introduces a novel definition of spectral subtriple that generalizes submanifolds within the spectral triple framework, extending existing notions of morphisms.

Findings

Definitions work in relevant almost commutative examples

Proposes conditions on Dirac operators for isometric submanifolds

Establishes a framework for noncommutative submanifolds

Abstract

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples.