Geodesic
On the duality in the theory of smooth manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 132-136
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In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings $f\in \operatorname{Hom}(M,M')$ of a smooth real finite-dimensional manifold $M$ into a similar manifold $M'$ and the family of homomorphisms $\varphi$ of the algebra $C^{\infty}(M')$ of smooth scalar-valued functions on $M'$ into the analogous algebra $C^{\infty}(M)$ on $M$, $\varphi\in \operatorname{Hom}\big(C^{\infty}(M'),C^{\infty}(M)\big)$. This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form.
Keywords: smooth manifold, smooth function, duality.
Mots-clés : homomorphism 
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     title = {On the duality in the theory of smooth manifolds},
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A. V. Ovchinnikov. On the duality in the theory of smooth manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 132-136. http://geodesic.mathdoc.fr/item/INTO_2020_185_a9/

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