Substituting (5) and (7) into (4) we can get a smooth MAP
a smooth map
. A point p [member of] [M.sup.n] is called a singular point if f is not an immersion at p.
A Hom-Lie algebroid is a quintuple (A, [phi], [x, x], [[rho].sub.A], [[alpha].sub.A]), where A is a vector bundle over a manifold M, [phi]: M [right arrow] M is a smooth map
, [x, x] : [GAMMA](A) [cross product] [GAMMA](A) [right arrow] [GAMMA](A) is a bilinear map, called bracket, [[rho].sub.A] : [GAMMA](A) [right arrow] [phi]'TM is a vector bundle morphism, called anchor, and [[alpha].sub.A] : [GAMMA](A) [right arrow] [GAMMA](A) is a linear endomorphism of [GAMMA](A), for X, Y [member of] [GAMMA](A), f [member of] [C.sup.[infinity]](M) such that
Consider a smooth map
[PHI] : Q x N [right arrow] P.
By a (smooth) variation of f we mean a smooth map
: M x (-[epsilon], [epsilon]) [right arrow] N, (x, t) [right arrow] [f.sub.t](x) ([epsilon] > 0) such that [f.sub.0] = f.
Let (M, g) and (N, h) be Lorentzian manifolds and [phi] : M [right arrow] N a smooth map
. Denote by [[nabla].sup.[phi]] the connection of the vector bundle [[phi].sup.*] TN induced from the Levi-Civita connection [[nabla].sup.h] of (N,h) .
In particular, in  Day and Shafer argued that the trapping set of their nonlinear business cycle model in one of the most interesting cases is well approximated by a piecewise smooth map
with two turning points with dynamics bounded in an absorbing interval.
A completely integrable system (or simply an integrable system) on a 2n-dimensional symplectic manifold (M, [omega]) is a smooth map
F := ([f.sub.1], ..., [f.sub.n]) : M [right arrow] [R.sup.n] such that each [f.sub.i] is constant along the flow (1) of each Hamiltonian vector field [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by Hamilton's equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and, moreover, the vector fields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are linearly independent almost everywhere on M.
For every local diffeomorphism f : M [right arrow] M and every smooth map
g : N [right arrow] [bar.N] we define
Recall that a smooth map
[phi] : [OMEGA] [right arrow] [H.sup.2n+1] is said to be harmonic if it is a critical point of the energy functional
Let (M,g) be an n-dimensional Riemannian manifold and [SIGMA] [subset] M be a Riemannian surface and [zeta] : [SIGMA] [right arrow] M a smooth map
. The pull-back bundle [[zeta].sup.*] (TM) has a metric and compatible connection, the pull-back connection, induced by the Riemannian metric and the Levi-Civita connection of M.
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