On Invariant Operations on a Manifold with a Linear Connection and an Orientation
Abstract
:1. Introduction
2. The Category of Ringed Spaces
2.1. Limits of Ringed Spaces
- -
- The underlying topological space is the inverse limit of the topological spaces , i.e., the set
- -
- Its sheaf of smooth functions is the direct limit .
2.2. Quotients by the Action of Groups
2.3. Differential Operators
- 1.
- is an open set of .
- 2.
- The map , defined as , is smooth.
3. Natural Operations in the Presence of an Orientation
3.1. Natural Bundles
- 1.
- Functorial character: and .
- 2.
- Local character: For any diffeomorphism and any open subset ,
- 3.
- Regularity: If is a smooth family of diffeomorphisms between open sets on X, then the family is also smooth.
3.2. Natural Operations in the Presence of an Orientation
4. Invariants of Linear Connections and an Orientation
- 1.
- They are symmetric in the last m covariant indices:
- 2.
- The symmetrization of the covariant indices is zero:
5. An Application
5.1. Invariant Theory of the Special Linear Group
- There exists a linear isomorphism .
- If is a sub-representation, then any equivariant linear map is the restriction of an equivariant linear map .
5.2. Uniqueness of the Torsion and Curvature Operators
5.3. Vector-Valued Natural Forms
5.4. Endomorphism-Valued Natural Forms
- (a)
- ,
- (b)
- ,
- (c)
- ,
- (d)
- .
Author Contributions
Funding
Conflicts of Interest
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Gordillo-Merino, A.; Martínez-Bohórquez, R.; Navarro-Garmendia, J. On Invariant Operations on a Manifold with a Linear Connection and an Orientation. Mathematics 2021, 9, 2577. https://doi.org/10.3390/math9202577
Gordillo-Merino A, Martínez-Bohórquez R, Navarro-Garmendia J. On Invariant Operations on a Manifold with a Linear Connection and an Orientation. Mathematics. 2021; 9(20):2577. https://doi.org/10.3390/math9202577
Chicago/Turabian StyleGordillo-Merino, Adrián, Raúl Martínez-Bohórquez, and José Navarro-Garmendia. 2021. "On Invariant Operations on a Manifold with a Linear Connection and an Orientation" Mathematics 9, no. 20: 2577. https://doi.org/10.3390/math9202577
APA StyleGordillo-Merino, A., Martínez-Bohórquez, R., & Navarro-Garmendia, J. (2021). On Invariant Operations on a Manifold with a Linear Connection and an Orientation. Mathematics, 9(20), 2577. https://doi.org/10.3390/math9202577