Isometric Spectral Subtriples
Paolo Bertozzini, Wanchalerm Sucpikarnon, Apimook Watcharangkool

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.
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