On Invariant Operations on a Manifold with a Linear Connection and an Orientation

We prove a theorem that describes all possible tensor-valued natural operations in the presence of a linear connection and an orientation in terms of certain linear representations of the special linear group. As an application of this result, we prove a characterization of the torsion and curvature...

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Published in:Mathematics (Basel) 2021-10, Vol.9 (20), p.2577
Main Authors: Gordillo-Merino, Adrián, Martínez-Bohórquez, Raúl, Navarro-Garmendia, José
Format: Article
Language:English
Subjects:
curvature operator
Geometry
linear connections
Mathematical functions
Mathematics
natural tensors
Neighborhoods
Operators
Orbits
Quantum field theory
Tensors
Theory of relativity
torsion tensor
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description We prove a theorem that describes all possible tensor-valued natural operations in the presence of a linear connection and an orientation in terms of certain linear representations of the special linear group. As an application of this result, we prove a characterization of the torsion and curvature operators as the only natural operators that satisfy the Bianchi identities.
doi_str_mv 10.3390/math9202577
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subjects curvature operator
Geometry
linear connections
Mathematical functions
Mathematics
natural tensors
Neighborhoods
Operators
Orbits
Quantum field theory
Tensors
Theory of relativity
torsion tensor
title On Invariant Operations on a Manifold with a Linear Connection and an Orientation
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