[2] **viXra:1410.0160 [pdf]**
*replaced on 2015-02-18 05:44:42*

**Authors:** Wenceslao Segura González

**Comments:** 46 Pages. Spanish

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.

**Category:** Geometry

[1] **viXra:1410.0139 [pdf]**
*submitted on 2014-10-22 19:49:20*

**Authors:** Linfan MAO

**Comments:** 25 Pages.

As we known, an objective thing not moves
with one's volition, which implies that all contradictions,
particularly, in these semiotic systems for things are artificial.
In classical view, a contradictory system is meaningless, contrast
to that of geometry on figures of things catched by eyes of human
beings. The main objective of sciences is holding the global
behavior of things, which needs one knowing both of compatible and
contradictory systems on things. Usually, a mathematical system
including contradictions is said to be a {\it Smarandache system}.
Beginning from a famous fable, i.e., the $6$ blind men with an
elephant, this report shows the geometry on contradictory systems,
including non-solvable algebraic linear or homogenous equations,
non-solvable ordinary differential equations and non-solvable
partial differential equations, classify such systems and
characterize their global behaviors by combinatorial geometry,
particularly, the global stability of non-solvable differential
equations. Applications of such systems to other sciences, such as
those of gravitational fields, ecologically industrial systems can
be also found in this report. All of these discussions show that a
non-solvable system is nothing else but a system underlying a
topological graph $G\not\simeq K_n$, or $\simeq K_n$ without common
intersection, contrast to those of solvable systems underlying $K_n$
being with common non-empty intersections, where $n$ is the number
of equations in this system. However, if we stand on a geometrical
viewpoint, they are compatible and both of them are meaningful for
human beings.

**Category:** Geometry