On some extension of the second order tangent space for a smooth manifold
- DOI
- 10.5922/0321-4796-2022-53-9
- Pages
- 94-111
Abstract
This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold.
Using the perturbation of the exterior derivative and ordinary differential, mappings are introduced that enable us to construct non-symmetrical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimensional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold.
A deformed external differential is widely used, which is a differential, i. e., its reapplication vanishes. We introduce a deformed external differential being a differential along the curves on the manifold, i. e., its repeated application along the curves on the manifold gives zero.
Reference
1. Henniart, G.: Les inégalités de Morse. Séminaire Bourbaki, exp. no 617, Astérisque, t. 121—122, 43—61 (1985).
2. Sulanke, R., Wintgen, P.: Differentialgeometrie und faserbundel. Basel (1972).
3. Laptev, G. F.: Fundamental infinitesimal structures of higher orders on a smooth manifold. Tr. Geom. Sem., 1, 139—189 (1966).
4. Lumiste, Yu. G.: Connections in homogeneous bundles. Sb. Math., 69, 419—454 (1966).
5. Petrova, L. I.: Skew-symmetric differential forms: Conservation laws. Fundamentals of field theory. Moscow (2006).
6. Polyakova, K.: Generalization of exterior differential by means of virtual function. DGMF. Kaliningrad. 41, 111—117 (2010).
7. Polyakova, K. V.: Second-Order Tangent-Valued Forms. Math. Notes, 105:1, 71—79 (2019).
8. Rybnikov, A. K.: Affine connections of second order. Math. Notes, 29:2, 143—149 (1981).
9. Rybnikov, A. K.: Second-order generalized affine connections. Izvestia Vuzov. Math., 27:1, 84—93 (1983).
10. Solodov, N. V.: Bivariant cohomology with symmetries. PhD thesis. Moscow, 2003.
11. Ho, F.-H.: Witten Deformation and Its Application toward Morse Inequalities. arXiv:1710.09579v1.
12. Petrova, L.: Evolutionary Relation of Mathematical Physics Equations Evolutionary Relation as Foundation of Field Theory Interpretation of the Einstein Equation. Axioms, 10:46 (2021). https://doi.org/ 10.3390/axioms10020046.
13. Petrova, L. I.: Skew-symmetric differential forms. Conservation laws: The foundation of equations of mathematical physics and field theory. Moscow (2021).
14. Polyakova, K. V.: Prolongations generated by horizontal vectors. J. Geom., 110:53 (2019). https://doi.org/10.1007/s00022-019-0510-2.
15. Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom. 17:4, 661—692 (1982).
16. Witten, E.: A new look at the path integral of quantum mechanics. arXiv:1009.6032v1.