Ninul A. S. Tensor Trigonometry. – Moscow: Scientific Publisher “Fizmatlit”, 2021, 320 p., 8 ill. (in English).
See this book, for example, in the Internet Archive page
https://archive.org/details/tensor-trigonometry-by-ninul-a-s-moscow-fizmatlit-2021-320-p-eng
or in the Google Books page
https://books.google.ru/books?id=0mceEAAAQBAJ

Description
The main goals of this monograph are (as 1st step) to develop a number of algebraic and geometric notions in Theory of Exact Matrices (Part I, Chapters 1-4), and then (as 2nd step) to work out on the gotten platform the basic contents of Tensor Trigonometry with bivalent tensor angles formed by two linear subspaces in the superspace or formed by rotation of a linear subspace in the superspace (Part II, Chapters 5-12). Since Tensor Trigonometry has a lot of applications in other mathematical and some physical domains, the largest examples of them are exposed in the book’s large Appendix.
Planimetry includes metric part and trigonometry. In geometries of metric spaces, from the end of XIX age, their tensor forms are widely used. However the flat trigonometry is remained only in its scalar forms in a plane or in a pseudoplane. The tensor trigonometry is development of the flat scalar trigonometry from Leonard Euler classic forms into multi-dimensional tensor forms (at n ≥ 2) with their vector and scalar orthoprojections in an admissible coordinate base, and with step by step increasing a complexity and opportunities. Described in the book are fundamentals of this new math subject with many various examples of its applications.
In theoretic plan, the tensor trigonometry with its binary tensor angles complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives the clear instrument for solutions of various geometric and physical problems in homogeneous isotropic spaces, such as Euclidean, quasi-Euclidean and pseudo-Euclidean ones (i.e., spaces with quadratic metrics); and also in non-Euclidean spherical or hyperbolic subspaces of constant radius-parameter R embedded into them. For these spaces and subspaces, the elementary kinds of the tensor trigonometry give very clearly general laws of summing rotations (motions) in complete forms with their polar decompositions into principal (spherical or hyperbolic) and secondary (orthospherical) ones. Besides, it gives all various limited descriptive trigonometric vector models of multi-dimensional non-Euclidean geometries on the projective plane or cylinder and many of applications in other domains of mathematics and physics. So, for non-collinear pseudo-Euclidean motions, a new interpretation of the orthospherical Thomas precession is given as manifestation of the induced Coriolis acceleration in the Minkowski space-time. In Theory of relativity, these applications were considered till a complete tensor trigonometric 4D pseudoanalog for world lines of the classic 3D theory by Frenet–Serret for Euclidean curves with all absolute and relative differentially-geometric, kinematic and dynamic characteristics in current world points of a world line.
The book is intended for researchers in the fields of multi-dimensional spaces, analytic geometry, linear algebra with theory of exact matrices, non-Euclidean geometries, theory of relativity, and also to all those who is interested in new knowledges and applications, given by exact sciences. It may be useful for educational purposes on this new math subject in the university departments of algebra, geometry and physics.
In the Kunstkammer at this book end, the readers may test themselves in solving a number of the suggested by the author questions and tasks near to the work’s topics.
In a paper form, without having this book, one may read it, for example, in the largest Russian State Library. In a digital form one may read or upload it, for instance, in the Internet Archive or in the Google Books (see web-address above), also in E-Books Directory (section Tensor Calculus), in E-library.ru, e-library of MSU’s Mech-Math Faculty (section Geometry and Topology), etc.
ISBN 978-5-94052-278-2, DOI 10.32986/978-5-94052-278-2-320-01-2021, Open Library ID Number: OL35374290M.
All rights reserved. Copyright: © 2021 by Anatoly S. Ninul
Say also (!), that the algebraic part of investigations (concerning theory and solution of algebraic equations), considered in the beginning of Chapter 1 along the way, was subsequently brought by the author to its full logical development and conclusion in his next math monograph in the frame of another item:
Ninul A. S. Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments. – Moscow, Fizmatlit, 2009, 336 p.
See this edition, for example, in the Internet Archive page
https://archive.org/details/optimizatsija-tselevykh-funktsij-by-ninul-a-s-moscow-fizmatlit-2009-336-p
or in the Google Books page
https://books.google.ru/books/?id=2PQuEAAAQBAJ
This book is filling up also the existing "blind spots" in the important math field as Optimization. But the book is so far only in Russian language. The author will welcome this book's initiative translation into English too by a specialist with interest to this very important mathematical field, even with the use of an advanced electronic translator, but with conservation all of its 600 formulas, 18 figures and diagrams, and 5 tables.
Personal author's web-site for communications: http://ninul-eng.narod.ru