Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Synthetic Geometry of Manifolds
  • Cited by 9
Publisher:
Cambridge University Press
Online publication date:
May 2010
Print publication year:
2009
Online ISBN:
9780511691690
DOI:
https://doi.org/10.1017/CBO9780511691690
Subjects:
Mathematics (general), Mathematics, Physics and Astronomy, Theoretical Physics and Mathematical Physics, Algebra
Series:
Cambridge Tracts in Mathematics (180)
Information

Book description

This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.

Reviews

'The book would certainly make a good graduate textbook. it is clearly written and contains a reasonable number of nontrivial exercises.'

Source: Zentralblatt MATH

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
Bibliography
Ambrose, W. and Singer, I. M. (1953) A theorem on holonomy, Transactions of the American Mathematical Society 75, 428–443.
Baez, J. and Schreiber, U. (2006) Higher Gauge Theory, arXiv:math/0511710v2 [math.DG].
Barr, M. (1971) Exact categories, in Barr, M., Grillet, P. A. and Osdol, D. H. (eds), Exact Categories and Categories of Sheaves, Springer Lecture Notes in Mathematics, Vol. 236, pp. 1–120.
Bell, J. (1998) A Primer of Infinitesimal Analysis, Cambridge University Press.
Bergeron, F. (1980) Objet infinitésimal en géométrie différentielle synthétique, Exposé 10 in Rapport de Recherches du Dépt. de Math. et de Stat. 80-11 and 80-12, Université de Montréal.
Bishop, R. L. and Crittenden, R. J. (1964) Geometry of Manifolds, Academic Press.
Breen, L. and Messing, W. (2001) Combinatorial differential forms, Advances in Math. 164, 203–282.
Brown, R. and Higgins, P. J. (1981) On the algebra of cubes, Journal of Pure and Applied Algebra 21, 233–260.
Brown, R. and Spencer, C. (1976) Double groupoids and crossed modules, Cahiers de Topologie et Géométrie Différentielle 17, 343–362.
Bunge, M. and Dubuc, E. (1987) Local concepts in synthetic differential geometry and germ representability, in Kueker, E., Lopez-Escobar, E. G. K. and Smith, C. H. (eds), Mathematical Logic and Theoretical Computer Science, Marcel Dekker.
Burke, W. L. (1985) Applied Differential Geometry, Cambridge University Press.
Demazure, M. and Gabriel, P. (1970) Groupes AlgébriquesTome I, Masson and North Holland.
Dubuc, E. (1979) Sur les modèles de la géometrie différentielle synthétique, Cahiers de Topologie et Géométrie Différentielle 20, 231–279.
Dubuc, E. (1990) Germ representability and local integration of vector fields in a well adapted model of SDG, Journal of Pure and Applied Algebra 64, 131–144.
Dubuc, E. and Kock, A. (1984) On 1-form classifiers, Communications in Algebra 12, 1471–1531.
Ehresmann, C. (1954) Structures locales, Annali di Matematica, 133–142.
Ehresmann, C. (1950) Les connexions infinitésimales dans un espace fibré differentiable, Colloque de Topologie, Bruxelles, CBRM.
Faran, J. (1998) A synthetic Frobenius Theorem, Journal of Pure and Applied Algebra 128, 11–32.
Felix, Y. and Lavendhomme, R. (1990) On DeRham's theorem in synthetic differential geometry, Journal of Pure and Applied Algebra 65, 21–31.
Frölicher, A. and Kriegl, A. (1988) Linear Spaces and Differentiation Theory, Wiley-Interscience.
Grandis, M. (2001) Finite sets and symmetric simplicial sets, Theory and Applications of Categories 8, 244–252.
Grandis, M. and Mauri, L. (2003) Cubical sets and their site, Theory and Applications of Categories 11, 186–211.
Greub, W. (1978) Multilinear Algebra (2nd edition), Springer Universitext.
Grothendieck, A. (1967) Élements de Géométrie Algébrique IV, Étude locale de schémas et des morphismes de schémas, part 4, Publ. Math. 32, Bures-sur-Yvette.
Helgason, S. (1962) Differential Geometry and Symmetric Spaces, Academic Press.
Hilton, P. J. and Wylie, S. (1960) Homology Theory, Cambridge University Press.
Johnstone, P. T. (1977) Topos Theory, Academic Press.
Johnstone, P. T. (2002) Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides, Vols 43, 44, Oxford University Press.
Joyal, A. Structures Infinitésimales, Lecture, 30 March 1979 (handwritten notes by G. Reyes).
Joyal, A. and Moerdijk, I. (1994) A completeness theorem for open maps, Journal of Pure and Applied Logic 70, 51–58.
Klein, F. (1926) Vorlesungen über höhere Geometrie, Springer Verlag.
Kobayashi, S. and Nomizu, K. (1963) Foundations of Differential Geometry, Wiley New York.
Kock, A. (1977) A simple axiomatics for differentiation, Mathematica Scandinavica 40, 183–193.
Kock, A. (ed.) (1979) Topos Theoretic Methods in Geometry, Aarhus Mathematical Institute Various Publications Series No. 30.
Kock, A. (1980) Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle 21, 227–246.
Kock, A. (1981/2006) Synthetic Differential Geometry, LMS 51, Cambridge University Press (2nd edition, LMS 333, Cambridge University Press).
Kock, A. (1982a) Differential forms with values in groups, Bulletin of the Australian Mathematical Society 25, 357–386.
Kock, A. (1982b) The algebraic theory of moving frames, Cahiers de Topologie et Géométrie Différentielle 23, 347–362.
Kock, A. (ed.) (1983a) Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Mathematical Institute Various Publications Series No. 35.
Kock, A. (1983b) Some problems and results in synthetic functional analysis, in Kock, A. (ed.) Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Mathematical Institute Various Publications Series No. 35, pp. 168–191.
Kock, A. (1984) A combinatorial theory of connections, in Gray, J. (ed.) Mathematical Applications of Category Theory, Proceedings 1983, AMS Contemporary Mathematics Vol. 30, pp. 132–144.
Kock, A. (1985) Combinatorics of non-holonomous jets, Czechoslovak Mathematical Journal 35, 419–428.
Kock, A. (1986a) Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques 27, 3–17. Corrections in Kock and Reyes (1987).
Kock, A. (1986b) Introduction to synthetic differential geometry, and a synthetic theory of dislocations, in Lawvere, F. W. and Schanuel, S. (eds), Categories in Continuum Physics, Proceedings Buffalo 1982, Springer Lecture Notes Vol. 1174.
Kock, A. (1989) On the integration theorem for Lie groupoids, Czechoslovak Mathematical Journal 39, 423–431.
Kock, A. (1996) Combinatorics of curvature, and the Bianchi Identity, Theory and Applications of Categories 2, 69–89.
Kock, A. (1998) Geometric construction of the Levi–Civita parallelism, Theory and Applications of Categories 4, 195–207.
Kock, A. (2000) Differential forms as infinitesimal cochains, Journal of Pure and Applied Algebra 154, 257–264.
Kock, A. (2001) Infinitesimal aspects of the Laplace operator, Theory and Applications of Categories 9, 1–16.
Kock, A. (2003) First neighbourhood of the diagonal, and geometric distributions, Universitatis Iagellonicae Acta Mathematica 41, 307–318.
Kock, A. (2004) A geometric theory of harmonic and semi-conformal maps, Central European Journal of Mathematics 2, 708–724.
Kock, A. (2006) Connections and path connections in groupoids, Aarhus Mathematical Institute Preprint No. 10. http://www.imf.au.dk/publs?id=619.
Kock, A. (2007a) Envelopes – notion and definiteness, Beiträge zur Algebra und Geometrie 48, 345–350.
Kock, A. (2007b) Principal bundles, groupoids, and connections, in Kubarski, J., T.Pradines, , Rybicki, T. and Wolak, R. (eds), Geometry and Topology of Manifolds (The Mathematical Legacy of Charles Ehresmann), Banach Center Publications 76, pp. 185–200.
Kock, A. (2007c) Infinitesimal cubical structure, and higher connections, arXiv:0705.4406[math.CT]
Kock, A. (2008) Combinatorial differential forms – cubical formulation, Applied Categorical Structures DOI 10.1007/s10485-008-9143–6.
Kock, A. and Lavendhomme, R. (1984) Strong infinitesimal linearity, with applications to strong difference and affine connections, Cahiers de Topologie et Géometrie Differentielle Catégoriques 25, 311–324.
Kock, A. and Reyes, G. E. (1979a) Manifolds in formal differential geometry, in Applications of Sheaves, Proceedings Durham 1977, Springer Lecture Notes in Mathematics Vol. 753.
Kock, A. and Reyes, G. E. (1979b) Connections in formal differential geometry, in Topos Theoretic Methods in Geometry, Aarhus Mathematical Institute Various Publications Series No. 30.
Kock, A. and Reyes, G. E. (1987) Corrigendum and addenda to “Convenient vector spaces embed”, Cahiers de Topologie et Géométrie Différentielle Catégoriques 28, 99–110.
Kock, A. and Reyes, G. E. (2003) Some calculus with extensive quantities: wave equation, Theory and Applications of Categories 11, 321–336.
Kock, A. and Reyes, G. E. (2006) Distributions and heat equation in SDG, Cahiers de Topologie et Géométrie Différentielle Catégoriques 47, 2–28.
Kock, A., Reyes, G. E. and Veit, B. (1979/80) Forms and integration in synthetic differential geometry, Aarhus Preprint Series No. 31.
Kolar, I. (1982) On the second tangent bundle and generalized Lie derivatives, Tensor N.S. 38, 98–102.
Kriegl, A. and Michor, P. (1997) The Convenient Setting of Global Analysis, American Mathematical Society.
Kumpera, A. and Spencer, D. (1973) Lie Equations. Volume I: General Theory, Annals of Mathematics Studies Number 73, Princeton University Press.
Lambek, J. and Scott, P. (1986) Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press.
Lavendhomme, R. (1987) Lecons de géométrie différentielle synthétique naïve, CIACO, Louvain-la-Neuve.
Lavendhomme, R. (1994) Algèbres de Lie et groupes microlinéaires, Cahiers de Topologie et Géométrie Différentielle Catégoriques 35, 29–47.
Lavendhomme, R. (1996) Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers.
Lawvere, F. W. (1979) Categorical dynamics, in Topos Theoretic Methods in Geometry, Aarhus Mathematical Institute Various Publications Series No. 30, pp. 1–28.
Lawvere, F. W. (1998) Outline of Synthetic Differential Geometry, 14 pp. Unpublished manuscript, www.acsu.buffalo.edu.wlawvere/downloadlist.html.
Lawvere, F. W. (2000) Comments on the development of topos theory, in Bier, J.-P. (ed.), Development of Mathematics 1950–2000, Birkhäuser Verlag, Basel, pp. 715–734.
Lawvere, F. W. (2002) Categorical algebra for continuum micro physics, Journal of Pure and Applied Algebra 175, 267–287.
Lawvere, F. W., Maurer, C. and Wraith, G. C. (eds) (1975) Model Theory and Topoi, Springer Lecture Notes in Mathematics Vol. 445.
Libermann, P. (1971) Sur les prolongements des fibrés principaux et des groupoïdes différentiables banachiques, in Analyse Globale, Séminaire de Mathématiques Supérieures, Les Presses de l'Université de Montréal, pp. 7–108.
Lie, S. (1896/1977) Geometrie der Berührungstransformationen, Leipzig 1896 (reprinted by Chelsea Publishing Company, 1977).
Mac Lane, S. (1971) Categories for the Working Mathematician, Springer Graduate Texts in Mathematics No. 5, Springer Verlag.
Mac Lane, S. and Moerdijk, I. (1992) Sheaves in Geometry and Logic, Springer Universitext.
Mackenzie, K. C. H. (1987) Lie Groupoids and Lie Algebras in Differential Geometry, LMS 124, Cambridge University Press.
Mackenzie, K. C. H. (1995) Lie algebroids and Lie pseudoalgebras, Bulletin of the London Mathematical Society 27, 97–147.
Madsen, I. and Tornehave, J. (1997) From Calculus to Cohomology, Cambridge University Press.
Malgrange, B. (1972) Equations de Lie, I, Journal of Differential Geometry 6, 503–522.
McLarty, C. (1983) Local, and some global, results in synthetic differential geometry, in Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Mathematical Institute Various Publications Series No. 35, pp. 226–256.
McLarty, C. (1995) Elementary Categories, Elementary Toposes, Oxford Logic Guides 21, Clarendon Press, Oxford.
Meloni, G.-C. and Rogora, E. (1988) Global and infinitesimal observables, in Borceux, F. (ed.), Categorical Algebra and its Applications, Proceedings Louvainla-Neuve 1987, Springer Lecture Notes 1348, pp. 270–279.
Minguez Herrero, C. (1988) Wedge products of forms in synthetic differential geometry, Cahiers de Topologie et Géométrie Différentielle Catégoriques 29, 59–66.
Moerdijk, I. and Reyes, G. E. (1991) Models for Smooth Infinitesimal Analysis, Springer.
Mumford, D. (1965/1988) Introduction to algebraic geometry (Preliminary version of first three chapters), Harvard notes 1965, (reprinted in Mumford, D., The Red Book of Varieties and Schemes, Springer Lecture Notes in Mathematics 1358, 1988).
Nelson, E. (1967) Tensor Analysis, Princeton University Press.
Nishimura, H. (1997) Theory of microcubes, International Journal of Theoretical Physics 36, 1099–1131.
Nishimura, H. (1998) Nonlinear connections in synthetic differential geometry, Journal of Pure and Applied Algebra 131, 49–77.
Nishimura, H. (2004) Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie 45, 677–696.
Nishimura, H. (2007) The Lie algebra of the group of bisections, Far East Journal of Mathematical Sciences 24, 329–342.
Nishimura, H. (2008a) The Frölicher–Nijenhuis Calculus in Synthetic Differential Geometry, arXiv:0810.5492[math.DG]
Nishimura, H. (2008b) Curvature in synthetic differential geometry of groupoids, Beiträge zur Algebra und Geometrie 49, 369–381.
Noll, W. (1967) Materially uniform simple bodies with inhomogeneities, Archive for Rational Mechanics and Analysis 27, 1–32.
Palais, R.et al. (1965) Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematical Studies 57.
Penon, J. (1985) De l'infinitésimal au local, These de Doctorat d'Etat, Paris 7.
Pradines, J. (1967) Theorie de Lie pour les groupoides differentiables, C. R. Academy Paris 266, 245–248.
Reyes, G. and Wraith, G. C. (1978) A note on tangent bundles in a category with a ring object, Mathematica Scandinavica 42, 53–63.
Saunders, D. J. (1989) The Geometry of Jet Bundles, LMS 142, Cambridge University Press.
Schreiber, U. and Waldorf, K. (2008), Smooth Functors vs. Differential Forms, arXiv:0802.0663v2[mathDG]
Serre, J.-P. (1965) Lie Algebras and Lie Groups, Benjamin Publishing Company.
Spivak, M. (1979) A Comprehensive Introduction to Differential Geometry (Vols 1–5), Publish or Perish, Inc.
Virsik, J. (1971) On the holonomity of higher order connections, Cahiers de Topologie et Géométrie Différentielle 12, 197–212.
White, J. E. (1982) The Method of Iterated Tangents with Applications to Local Riemannian Geometry, Pitman Press.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.