Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus
Round 1
Reviewer 1 Report
Please see the attachment file.
Comments for author File: 
Comments.pdf
Author Response
Thank you, I've corrected the misspellings.
I wrote this paper as part of my PhD dissertation work at UC Santa Cruz in 2012, and only posted a pre-print of it on arxiv.org, but never got around to publishing it formally. It's correct to say there have been no revisions to it since 2012, and I think the work still stands as current and relevant.
I'm an independent mathematician, and don't have an associated institution, nor an institutional email address. I do software engineering for money, and academic math research on my own time.
Regarding formatting using the MDPI template, I believe I should be able to upload a new revision of the manuscript after I've replied to all of the reviewer comments.
Reviewer 2 Report
In this paper, the notion of strongly typed language have been used to characterize the field of computer programming and also to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, is also used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to accomodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds is formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler-Lagrange equations, and the second variation of such functionals is derived. The paper contains new and interesting results, also contains some important results regarding the variations calculus and also the theory of manifolds.
The computations along the paper are done properly and the paper contains new and interesting results. The paper could be published as it is.
Author Response
Thank you.
