Model of hyperbolic geometry of Lobachevsky geometry

Riemannian geometry takes Euclidean geometry and various non-Euclidean geometries as special cases. For example, if you define a metric (a is a constant), then when a 0 is ordinary Euclidean geometry, when a 0 is elliptical geometry, and when a 0 is hyperbolic geometry (Lobachevsky geometry).

Riemann sees the surface itself as a separate geometric entity, rather than as a mere geometric entity in Euclidean space. He first developed the concept of space, proposing that the object of geometry should be a multiplicity generalized quantity, and that the points in space can be n real numbers (x1,......xn) as coordinates. This is the original form of the modern n-dimensional differential manifold, which laid the foundation for describing natural phenomena in abstract space.

This spatial geometry should be based on infinitely adjacent two points (x1, x2,......xn) and (x1 dx1,......xn dxn), measured by the positive definite quadratic form determined by the square of the length of the differential arc. That is, (gij) is a positively definite symmetric matrix composed of functions. This is the Riemann metric.

Riemann recognized that a metric is just a structure added to a manifold and that there can be many different measures on the same manifold. Mathematicians before Riemann only knew that there was an induced metric ds2 edu2 2fdudv gdv2 on the surface s in the three-dimensional Euclidean space e3, i.e., the first elementary form, but did not realize that s could also have a metric structure independent of the three-dimensional Euclidean geometry. Riemann realized the importance of distinguishing between induced and independent Riemannian metrics, and thus freed himself from the constraints of classical differential geometry surface theory that was limited to induced metrics, and founded Riemannian geometry.

Geometric propositions that do not involve the axioms of parallelism are also true in Euclidean geometry as they are true in hyperbolic geometry. Propositions that rely on the axioms of parallelism do not hold true in hyperbolic geometry. Here are a few examples:

Euclidean geometry: Perpendicular and diagonal lines of the same straight line intersect.

Two lines perpendicular to the same line are parallel.

There are similar but not congruent polygons.

Crossing three points that are not on the same line can be done and can only be made in a circle.

Hyperbolic geometry: Perpendicular and diagonal lines of the same line do not necessarily intersect.

Two straight lines perpendicular to the same straight line, when extended at both ends, are discrete to infinity. There are no similar but not congruent polygons.

Crossing three points that are not on the same straight line does not necessarily make a circle.

As can be seen from some of the propositions of Drochevsky's geometry listed above, these propositions contradict the intuition to which we are accustomed. So some of the geometric facts in Lobachevsky geometry are not as easily accepted as Euclidean geometry. However, it is correct to interpret Roche geometry by making an intuitive "model" of the facts in Euclidean geometry that we are accustomed to.

Hyperbolic geometry, also known as Lobachevsky geometry, Polya-Lobachevsky geometry, or Roche geometry, is a geometric axiom system that is independent of Euclidean geometry. The axiomatic system of hyperbolic geometry differs from the axiom system of Euclidean geometry in that the "fifth axiom of Euclidean geometry" (also known as the axiom of parallelism, equivalent to "there is only one straight line parallel to a known line at a point outside the straight line") is replaced by the "hyperbolic parallel axiom" (equivalent to "at least two straight lines parallel to the known straight line at a point outside the straight line"). In this system of axioms, a series of new geometric propositions that differ from the content of Euclidean geometry can be proved through deductive reasoning, such as the inner angles of a triangle and less than 180 degrees.

Lobachevsky geometry is a non-European sock geometry. Silver scramble ().

a.That's right. b.Mistake.

Correct Answer: a