doi: 10.3934/jgm.2022007
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. DragerJ. M. LeeE. 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. JotzT. 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.

show all references

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. DragerJ. M. LeeE. 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. JotzT. 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.

[1]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301

[2]

Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

[3]

Françoise Demengel, Thomas Dumas. Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian". Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1135-1155. doi: 10.3934/dcds.2019048

[4]

L. Bakker, G. Conner. A class of generalized symmetries of smooth flows. Communications on Pure and Applied Analysis, 2004, 3 (2) : 183-195. doi: 10.3934/cpaa.2004.3.183

[5]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[6]

Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure and Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019

[7]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure and Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

[8]

Tao Chen, Linda Keen. Slices of parameter spaces of generalized Nevanlinna functions. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5659-5681. doi: 10.3934/dcds.2019248

[9]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043

[10]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675

[11]

Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072

[12]

Limin Wen, Xianyi Wu, Xiaobing Zhao. The credibility premiums under generalized weighted loss functions. Journal of Industrial and Management Optimization, 2009, 5 (4) : 893-910. doi: 10.3934/jimo.2009.5.893

[13]

Mihaela Roxana Nicolai, Dan Tiba. Implicit functions and parametrizations in dimension three: Generalized solutions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2701-2710. doi: 10.3934/dcds.2015.35.2701

[14]

Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3525-3550. doi: 10.3934/jimo.2020131

[15]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002

[16]

Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, 2022, 16 (4) : 753-769. doi: 10.3934/ipi.2021074

[17]

Boling Guo, Haiyang Huang. Smooth solution of the generalized system of ferro-magnetic chain. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 729-740. doi: 10.3934/dcds.1999.5.729

[18]

Qiyuan Wei, Liwei Zhang. An accelerated differential equation system for generalized equations. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021195

[19]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030

[20]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615

2021 Impact Factor: 0.737

Article outline

[Back to Top]