Extensions in affine spaces of bilinear applications, differentiable actions, and tensors
DOI:
https://doi.org/10.17268/sel.mat.2024.01.04Keywords:
Affine bilinear, affine action, affine tensorAbstract
In this article, several generalizations in affine spaces are studied. First, the notion of affine mappings to bilinear mappings defined in affine spaces is explored, referred to as affine bilinear mappings. Subsequently, differentiable actions of a Lie group on affine spaces are defined, and their isotropy group, orbit space, and set of fixed points are examined. Finally, the concept of tensor product between vector spaces is extended to the tensor product between affine spaces.
References
Castellenet M, Llerena I. Álgebra lineal y geometría. Universidad Autónoma de Barcelona: Reverté, S. A.; 2020.
Erdman JM. Elements of linear and multilinear algebra. Portlang State University, USA: World Scientific Publishing Company; 2021.
Delgado M. Geometría afín y Euclídea. Geometría afín y Euclídea: Departamento de Matemática Fundamental; 2012.
Lluis-Puebla E. Álgebra lineal, Álgebra Multilineal, y k-Teoría algebraica clásica. 2nd. ed. Universidad Nacional Autónoma de M´exico: Sociedad Matemática Mexicana; 2008.
Tarida AR. Affine maps, Euclidean motions and quadrics. New York: Springer-Verlag London; 2011.
Roman S. Advanced linear algebra. 3rd. ed. USA: Springer-Board; 2008.
Casas-Alvero E. Analitic proyective geometry. Universidad de Barcelona: European Mathematical Society; 2010.
López F. Geometría III. Universidad Granada: Departamento de Geometría y Topología; 2021.
Sancho De Salas J. The Fundamental Theorem of Affine and Projective Geometries. Preprint. 2022.
Sancho De Salas J. The Fundamental Theorem of Affine Geometry. Extracta Mathematicae. 2023;38(2):221-35.
Casimiro A, Rodrigo C. The Fundamental Theorem of Affine and Projective Geometries. Preprint. Mayo 2023.
Duretíc, J. From differentiation in affine spaces to connetions. Teaching of Mathematics.
;18(2):61-88.
Cooperstein BN. Advanced linear algebra. 2nd. ed. University of Califormia: CRC Press; 2015.
Bischop RL, Goldberg SI. Tensor analysis on manifolds. New York: Curier Corporation INC; 2012.
Lie JM. Introduction to smooth manifolds. 2nd. ed. New York: Springer; 2013.
Greub W. Multilinear algebra. 2nd. ed. New York Heidelberg Berlin: Springer Verlang; 1978.
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