nLab
convenient manifold

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Manifolds and cobordisms

          Contents

          Idea

          A manifold, possibly infinite-dimensional, is called a convenient manifold – implicitly meaning: convenient for differential geometry – if it is modeled on a convenient vector space.

          One should note that this usage of the adjective ‘convenient’ is different to that in ‘convenient category’, for example of smooth spaces. In that case the category is convenient, whereas here the objects are convenient.

          Properties

          Embedding into the Cahiers-topos

          Together with convenient vector spaces, convenient manifods embed into the Cahier topos of synthetic differential smooth spaces. See at Cahiers topos for more on this.

          References

          A standard textbook reference is

          • Peter Michor, A convenient setting for differential geometry and global analysis, Cahiers Topologie Géom. Différentielle 25 (1984), no. 1, 63–109, MR86g:58014a; A convenient setting for differential geometry and global analysis. II, Cahiers Topologie Géom. Différentielle 25 (1984), no. 2, 113–178.

          A survey is for instance in the slides

          • Richard Blute, Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic (2011) (pdf)
          • John C. Baez, Alexander E. Hoffnung, Convenient categories of smooth spaces, Trans. Amer. Math. Soc. 363 (2011), 5789-5825 pdf

          Last revised on February 9, 2013 at 22:24:07. See the history of this page for a list of all contributions to it.