52,639 research outputs found
Differential Geometry from Differential Equations
We first show how, from the general 3rd order ODE of the form
z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on
the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s)
satisfies a special differential condition of the form, U[F]=0, the conformal
metric possesses a conformal Killing field, xi = partial with respect to s,
which in turn, allows the conformal metric to be mapped into a three
dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the
space of solutions of the original differential equation. This construction is
then generalized to the pair of differential equations, z_ss =
S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the
derivatives of z with respect to s and t. In this case, from S and T, one can
again, in a natural manner, construct a Lorentzian conformal metric on the six
dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations
analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then
possesses a pair of conformal Killing fields, xi =partial with respect to s and
eta =partial with respect to t which allows, via the mapping to the four-space
of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a
four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian
metrics can be constructed in this manner. This construction, with further
conditions on S and T, thus includes all (local) solutions of the Einstein
equations.Comment: 37 pages, revised version with clarification
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
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