nLab tangent map

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Given a differentiable map f:MNf:M\to N of differentiable manifolds, for each pMp \in M we have the differential df p:T pMT f(p)Nd f_p : T_p M \to T_{f(p)} N of ff at pp. These maps define a unique tangent map Tf:TMTNT f : T M \to T N of tangent bundles (as vector bundles) such that the diagram

T pM d pf T f(p)N TM Tf TN\array{ T_p M &\stackrel{d_p f}\to & T_{f(p)}N\\ \downarrow &&\downarrow \\ T M &\stackrel{T f}\to & T N }

commutes for each pp. The differential df pd f_p of ff at pp (or sometimes the composition T pMd pfT f(p)NTNT_p M\stackrel{d_p f}\rightarrow T_{f(p)}N\hookrightarrow T N) is then denoted T pfT_p f,

making TT into a functor (see differentiation).

Last revised on October 18, 2023 at 03:28:48. See the history of this page for a list of all contributions to it.