Heaps of Linear Connections and Their Endomorphism Truss

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author studies heap structure on the set of linear connections on a Lie algebroid.

They firstly methodically recall the notions of a semi-heap, and a truss and further the notion of the anchored vector bundle, linear connections on an anchored vector bundle and the ternary operation on the affine space of linear connections.

 

Further, they deal with the notions of metric compatible connections and show that they form an abelian subheap of the heap of auto-parallel sections.

Further some other consequences, such as the torsion and curvature of a triple product, Lie derivative of connections, dual connections,.. and their properties are presented. The author produces an explicit construction of the endomorphism truss of linear connections.

 

The subject of the paper is interesting and well set in the context of similar work recently published. The Introduction is informative and gives a nice motivation for the work that is going to be presented. In particular, I find the general descriptions of how some of the mathematical objects, which are later formally defined, should or could be understood, quite nice.

The results are presented in a clear and accurate manner.

Hence, I recommend this paper to be published in Symmetry.

 

Author Response

Dear Referee,

thank you for taking the time to read my paper and your comments.

Best,

Andrew

Reviewer 2 Report

Comments and Suggestions for Authors

Accepted after minor correction.

Comments for author File: Comments.pdf

Author Response

Dear Referee,

thank you for taking the time to read the paper and provide helpful comments. I have made all the changes suggested.  Specifically,

1. Added “ We remark that the use of ternary in differential geometry is novel and the endomorphism truss of linear connections provides a concrete geometric example of a truss.”

2. A paragraph on the arrangement of the paper has been added at the end of section 1.1

3. Changed to “role”.

4. Nakahara's book is cited for an introduction and overview of Chern-Weil theory.

5. Heisenberg (2024) added as a review of metric-affine theories (the paper contains a lot of citations to other works). Koyama (2018) has been added as an overview of gravity theories beyond general relativity.

6. Pradines (1974) is given as a reference for Definition 1. This is the earliest reference that I am aware of, though the notion is probably older.

7. The equation is now on a separate line.

8. Corrected.

9. Changed to et al.

10. It is explained that the projection is a smooth surjective submersion. Carmeli et al is cited as a place that details of what this means for supermanifolds can be found. The anchor has no further properties other than being a vector bundle homomorphism.

11. The tilde is the grading map, it gives 0 or 1 depending on the Grassmann parity of the objects. This is defined just before Example 1.

12. i=1,2,3 is added to definition 3.

13. It is now stated that the Gamma are the Christoffel symbols

14. The equations are now on separate lines.

15. Boucetta (2011) has been added as a reference for metric compatibility of connections on vector bundles – in this specific case Lie algebroids. The notion is probably a lot older, but I am unaware of specific references.
    

     

Once again, thank you for your comments.

Best,

Andrew

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is focused not on concrete results, it builds theory generalizing classical differential geometry. Author  examine the heap of linear connections on anchored vector bundles and Lie algebroids, naturally this covers the example of affine connections on a manifold. He presents some new interpretations of classical results via this ternary structure of connections. Endomorphism of linear connections are studied and their ternary structure, in particular the endomorphism truss, is explicitly presented. Some preliminary consequences, such as the torsion and curvature of a triple  product of connections have been presented. A key result here is the explicit construction  of the endomorphism truss of linear connections. He remarks that the results of this note extend verbatim to the algebraic setting of  (left) connections on anchored modules and Lie–Rinehart pairs over associative supercommutative, unital superalgebras. Indeed, we have avoided using local descriptions in any calculations. With a little effort, we expect the results presented here to generalise to the  setting of almost commutative Lie algebroids

It is nice paper. Author is famouse scientist with many publications in old reputational journals, including the paper subject, for example Bruce, A.J., Connections adapted to non-negatively graded structures, Int. J. Geom. Methods Mod. Phys. 16 (2019), no. 2, 1950021,  32 pp.  Bruce, A.J., Homological sections of Lie algebroids, Differ. Geom. Appl. 79 (2021) 101826.  Bruce, A.J., & Grabowski, J., Odd connections on supermanifolds: existence and relation with affine connections, J. Phys. A: Math.  Theor. 53 (2020) 455203. He works in very good university.

The paper is nice and I support it.

 

It is really good paer and I support it.

Author Response

Dear Referee,

thank you for taking the time to read my paper and for your comments.

Best,

Andrew