Chapter VI: The New Geometry Riemannian
Saturday, September 29, 2007
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Bernhard Riemann (1826-1866) was interested not only in two-dimensional flat spaces where they live in the triangle and the circle and three-dimensional space they live in the cube and sphere and even in the mathematical spaces four dimensions (hard to visualize but feasible to define and manipulate mathematically), but also was interested in much more general characterization of n-dimensional space, which makes the method more universal Riemannian The previously published works.
Its main contribution is in his presentation "Ueber die Hypothese, Welche der Geometrie zu Grunde liegen" (on the assumptions behind the Foundations of Geometry), lecture given in 1854 at the request of the Gauss, which leaves the Euclidean method of basing all demonstrations on which uses the rule and compass, redefining the geometry as the study of letters ( manifolds ) dimensional space (bounded ) and non-bounded (unbounded ) capable of containing any number of dimensions, together with a coordinate system, and a metric that defines the shortest distance between two points. In the three-dimensional Euclidean geometry, metrics, before he came Riemann, was inspired by the definition of an infinitesimal length ds line through the Cartesian coordinate system line as follows:
(ds) ² = (dx) ² + (dy) ² + (dz) ²
which has been the equivalent of (infinitesimal) of the Pythagorean Theorem. But Riemann generalized the concept of length through their letters, and these "letters" are what completely defines the space without any external frame of reference. We call it "differential geometry taken to the extreme." Before proceeding
talking about geometry generally developed by Riemann, is important to make a brief mention on that branch of mathematics whose principles were taken first by Gauss and later by Riemann to be expanded dramatically: the differential geometry. When this matter began to take shape, it had to start with something simpler study before this new field of study be generalized to more sophisticated things. And in this case, we had to begin the study of curves before climbing to the study of surfaces.
Previously, in Descartes invented analytic geometry by combining the Euclidean geometry and algebra, the concept of a curved line able to get out of the plane into three dimensions (known as warped curve) was something that through analytic geometry is used to describe doing something external reference to the curve, and the external that we are referring to a system coordinates (such as rectangular Cartesian coordinates x axes, and and z or polar coordinates, always referred to an origin point O external to the curve being described). For example, to describe a spiral curve (a curve in space with a spring) through a system of Cartesian coordinates in three dimensions, mental image we have of this description is as follows:
whose stroke in three dimensions is achieved through a system of equations as the following set of parametric equations :
y = RCOs ( t) z = Rsen(Giving various values \u200b\u200bto the variable t , my readers can check using a three dimensional coordinate system (x , and , z) that these equations are actually drawing a helical curve.)
(t) x = t
But like Euclid did not need analytic geometry to develop its flat geometry (in his time there was no algebra and had not even invented the zero), it was not long for mathematicians to realize that a curve could be described itself based on the local properties of the curve, without having to resort to an external reference. This means that an ant walking along the curve and carefully measuring the changes it encounters along his journey, without knowing anything about the existence of an external system of rectangular coordinates, to discover her own properties curve. Because what matters here are the changes to be experiencing curve, especially its infinitesimal changes to go forward ant arc differential increments ds along the curve, we are already talking about the beginnings of a geometry based on the tools of differential (infinitesimal), in other words, a differential geometry . Thus, if the concept of curvature a curved line at a given point of the line (defined as the inverse of the radius of the curve at that point) we add the concept of torque (with which curves admit they can get out of the plane by jumping into the Euclidean three dimensional space), and the independence of the concepts of curvature and torsion the necessity of having to rely on an external reference, then the torsion and curvature can give us all the information needed to describe the curve without external references. This was discovered independently by the mathematician Jean Frédéric Frenet (in 1847) and Joseph Alfred Serret (in 1851), which is immortalized in the Frenet-Serret relations :
that allow us to attach along a curve a trio (or triad ) rolling three vectors (a vector tangential to the curve always points along it, the vector T , a normal vector to the curve that always points toward the center to which scenario is shifting the curve, the vector N , and a vector B is always perpendicular to the vectors and T N ). Thus, in the new description of the helical curve:
The line, according to the geometry :
y = mx + b
The straight as differential geometry: Curvature
= 0
Torque = 0
Torque = 0
This tells us simply that the line is a line without bending or twisting , without having to make reference to the variables x and and Cartesian coordinates . (Another way of looking, taking into account the curvature is defined as the inverse of the radius of curvature, is assuming a straight line as a segment of a circle of radius of curvature is infinitely large, which effectively becomes a straight line.)
curve Now for the simplest of all:
circumference as analytic geometry :
x ² + y ² = R ²
circumference as differential geometry :
Curvature = 1 / R
Torque = 0
Torque = 0
This tells us that a circle is a geometric figure whose curvature is constant and is confined to two-dimensional plane (as the torque is zero and therefore is not a warped curve).
another example:
a helical curve, according to the geometry :
5cos
x = (t)a helical curve, according to the geometry Differential :
5sen
y = (t) z = t
Curvature Torsion = 0.5 = 2.5
As can be seen, the description of a helical curve by means of differential geometry is performed using not a set of three algebraic equations, but only two numbers which tell us that it is a warped curve is out of a plane (having a nonzero torsion), which do not have a twist would be simply a circle with a constant curvature of 0.5 (or, that is, a radius of curvature of 2 units, the curvature being defined as the inverse of the radius of curvature). The greater the torque of the helical curve, more "stretch" the spring will be like. Both
analytic geometry and differential geometry describing the same thing but in different ways, the first always makes using an external reference (Cartesian coordinates or polar coordinates, both with a point of origin) while the second makes no reference to an external reference.
course, no reason why we have to limit the study of differential geometry of space curves. We can extend the study of surfaces. And as we talk about the curvature of a line, we can speak of the curvature of a surface , the only difficulty lay in defining it so that the definition was accurate and useful. Thus was born the central idea behind the concept of Gaussian curvature . Riemann went even further than Gauss, and extended the concept of Gaussian curvature of a surface into a mathematical space of any dimension , and he did so it was possible to adapt not only Euclidean geometry, but even at the non-Euclidean geometry. Using as a resource
through the complex variable z
= x + iy
where x and and are variables real numbers and the number i equals the square root of -1, the metric Reimann for the Poincaré disk we've seen previously found to be (the proof is not difficult but requires the use of derivatives partial use of which may be unknown to most of my readers, why is omitted):
Note that for
x ² + y ² = 1
would correspond to the edge of the unit disk (radio = 1) Poincare metric ds becomes infinitely large (or, to use a nicer term for mathematicians, indefinite ), which agrees with the fact that the edge of the circle is the boundary line that represents the infinite within the investment drive.
The metric is what it tells us everything we want to know about the space we are studying. The metric tells us right away if space is Euclidean or non-Euclidean, all we have to do is to calculate the curvature of the surface, and if the surface has a curvature of zero, the space is Euclidean. Examples are traditional Euclidean space of two dimensions:
ds ² = dx ² + dy ²
or three-dimensional Euclidean space
ds ² = dx ² + dy ² + dz ²
or the space of four dimensions studied in the theory Special of Relativity:
ds ² = dx ² + dy ² + dz ² - c ² t ²
where space Einsteinian time-curve does not exhibit any surface. is in the General Theory of Relativity where this four dimensional space no longer has a zero curvature due to the presence of mass-energy that enters a curve that can be positive or negative, which tells us the metric Riemannian.
In general, for a three-dimensional space, the Riemann metrics to measure the distance (infinitesimal) between two points is as follows:
where the various coefficients can g constant and even functions of the variables (x , and , z) . The expansion into areas with a greater number of dimensions is carried out in the same way.
Thus, the curvature of space, for which Riemann also gave a definition, is defined entirely by the intrinsic properties of the cards in any space. In this new definition of geometry, geometry is reduced to sets of n-ples ( n-tuples ) ordered that combine according to certain rules. More important is the fact that the Riemannian curvature , traditional Euclidean geometry is the space defined by a constant Riemannian curvature zero, while the spherical geometry is defined by a Riemannian curvature of +1, the geometry entanto is defined by a hyperbolic Riemannian curvature -1. This marks the beginning of a new more general geometry of Euclid, as it includes all possible geometries, why Riemann was known as "the new Euclid."
If the Riemannian metric for a three-dimensional space we
and all other coefficients zero, then this metric reduces to
(ds) ² = (dx) ² + (dy) ² + (dz) ²
is, the formula to measure the distance (infinitesimal) ds along a line in a Euclidean space (no curvature) dimensional.
The Riemann metrics can be easily extended without problem to a four-dimensional space , in which case the expression for (ds ) ² will 16 terms instead of the nine we saw above. Below is this metric:
And it is precisely this Riemannian metric in a generalized space of four dimensions, which may be flat or curved, which Einstein used to construct his general theory Relativity.
In general, the metric for any number of dimensions is represented in a much more compact and easier to remember by using Einsteinian convention of repeated indices according to which, if in an abbreviated expression thus , there is a single index repeated two or more times, then it is understood to be carrying out a summation over that index:
As an example, for a four-dimensional space, the expansion of the subscript to that appears to us is repeated twice in the following four terms:
What Riemann did was extraordinary. He built a whole new way of generalized geometry, which includes all possible geometries (Euclidean, elliptic, hyperbolic), and left us a way (his "metrics") to measure distances within infinite varieties offered by the geometry that is consistent with the metrics used previously, and gave us a way of knowing, through the definition of curvature (the Riemann curvature tensor ), without having to go build figures and diagrams, if a geometry is specified by certain relations Euclidean, elliptic, or hyperbolic. The downside is that to achieve this victory to generalize, the mathematics required necessarily become complex (in-depth study on this issue would require going into detail on a topic known as tensor calculus, but in the modern approach is replaced the calculation tensor for something more current known as the exterior calculus or calculation of differential forms through the definition of "product key" or wedge product that brings with it a collection of axioms on which develops the topic .) The increase in the complexity brought by the new Riemannian geometry is inevitable. This is the price you pay when there is a leap of this nature. But considering the fact that the Riemannian geometry laid the groundwork for Einstein, invoking the concept of curved space in four dimensions, could develop his General Theory of Relativity (often called Geometrodinámica being essentially a geometric theory of gravity ), which has among other things correctly predicted the existence of black holes, the "gravitational lensing", and the expansion of the universe, the price paid by the complexity of Riemannian geometry is more than fair.
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