# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2022007
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## On embedding of subcartesian differential space and application

 School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received  January 2022 Early access April 2022

Consider a locally compact, second countable and connected subcartesian differential space with finite structural dimension. We prove that it admits embedding into a Euclidean space. The Whitney embedding theorem for smooth manifolds can be treated as a corollary of embedding for subcartesian differential space. As applications of our embedding theorem, we show that both smooth generalized distributions and smooth generalized subbundles of vector bundles on subcartesian spaces are globally finitely generated. We show that every algebra isomorphism between the associative algebras of all smooth functions on two subcartesian differential spaces is the pullback by a smooth diffeomorphism between these two spaces.

Citation: Qianqian Xia. On embedding of subcartesian differential space and application. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022007
##### References:
 [1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111. [2] M. Breuer and C. D. Marshall, Banachian differentiable spaces, Math. Ann., 237 (1978), 105-120.  doi: 10.1007/BF01351675. [3] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [4] K. Cegielka, Existence of smooth partition of unity and of scalar product in a differential space, Demonstratio Math., 6 (1973), 493-504. [5] R. Cushman, H. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems, World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2010. doi: 10.1142/7509. [6] R. Cushman and J. Śniatycki, Aronszajn and Sikorski subcartesian differential spaces, Bulletin Polish Acad. Sci. Math., 69 (2021), 171-180.  doi: 10.4064/ba201118-1-11. [7] L. D. Drager, J. M. Lee, E. Park and K. Richardson, Smooth distributions are finitely generated, Ann. Global Anal. Geom., 41 (2012), 357-369.  doi: 10.1007/s10455-011-9287-8. [8] J. Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. (Basel), 85 (2005), 190-196.  doi: 10.1007/s00013-005-1268-3. [9] M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. [10] M. Jotz, T. S. Ratiu and J. Sniatycki, Singular reduction of Dirac structures, Trans. Amer. Math. Soc., 363 (2011), 2967-3103.  doi: 10.1090/S0002-9947-2011-05220-7. [11] A. D. Lewis, Generalised subbundles and distributions: A comprehensive review, preprint, 2014. [12] A. D. Lewis, Gelfand duality for manifolds, and vector and other bundles, preprint, https://mast.queensu.ca/~andrew/papers/pdf/2020d.pdf, 2020. [13] C. D. Marshall, Calculus on subcartesian spaces, J. Differential Geom., 10 (1975), 551-573. [14] J. R. Munkres, Elementary Differential Topology, Princeton University Press, 1966. [15] J. P. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian Reduction, Birkhauser Boston, Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7. [16] R. Sikorski, Abstract covariant derivative, Colloq. Math., 18 (1967), 251-272.  doi: 10.4064/cm-18-1-251-272. [17] J. Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier, 53 (2003), 2257-2296.  doi: 10.5802/aif.2006. [18] J. Śniatycki, Generalizations of Frobenius' Theorem on manifolds and subcartesian spaces, Canad. Math. Bull., 50 (2007), 447-459.  doi: 10.4153/CMB-2007-044-2. [19] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139136990. [20] H. Whitney, Differentiable manifolds, Ann. of Math., 37 (1936), 645-680.  doi: 10.2307/1968482.

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##### References:
 [1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111. [2] M. Breuer and C. D. Marshall, Banachian differentiable spaces, Math. Ann., 237 (1978), 105-120.  doi: 10.1007/BF01351675. [3] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [4] K. Cegielka, Existence of smooth partition of unity and of scalar product in a differential space, Demonstratio Math., 6 (1973), 493-504. [5] R. Cushman, H. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems, World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2010. doi: 10.1142/7509. [6] R. Cushman and J. Śniatycki, Aronszajn and Sikorski subcartesian differential spaces, Bulletin Polish Acad. Sci. Math., 69 (2021), 171-180.  doi: 10.4064/ba201118-1-11. [7] L. D. Drager, J. M. Lee, E. Park and K. Richardson, Smooth distributions are finitely generated, Ann. Global Anal. Geom., 41 (2012), 357-369.  doi: 10.1007/s10455-011-9287-8. [8] J. Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. (Basel), 85 (2005), 190-196.  doi: 10.1007/s00013-005-1268-3. [9] M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. [10] M. Jotz, T. S. Ratiu and J. Sniatycki, Singular reduction of Dirac structures, Trans. Amer. Math. Soc., 363 (2011), 2967-3103.  doi: 10.1090/S0002-9947-2011-05220-7. [11] A. D. Lewis, Generalised subbundles and distributions: A comprehensive review, preprint, 2014. [12] A. D. Lewis, Gelfand duality for manifolds, and vector and other bundles, preprint, https://mast.queensu.ca/~andrew/papers/pdf/2020d.pdf, 2020. [13] C. D. Marshall, Calculus on subcartesian spaces, J. Differential Geom., 10 (1975), 551-573. [14] J. R. Munkres, Elementary Differential Topology, Princeton University Press, 1966. [15] J. P. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian Reduction, Birkhauser Boston, Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7. [16] R. Sikorski, Abstract covariant derivative, Colloq. Math., 18 (1967), 251-272.  doi: 10.4064/cm-18-1-251-272. [17] J. Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier, 53 (2003), 2257-2296.  doi: 10.5802/aif.2006. [18] J. Śniatycki, Generalizations of Frobenius' Theorem on manifolds and subcartesian spaces, Canad. Math. Bull., 50 (2007), 447-459.  doi: 10.4153/CMB-2007-044-2. [19] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139136990. [20] H. Whitney, Differentiable manifolds, Ann. of Math., 37 (1936), 645-680.  doi: 10.2307/1968482.
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