In the mathematical field of differential geometry , the Gauss–Bonnet theorem (or Gauss–Bonnet formula ) is a fundamental formula which links the curvature of a surface to its underlying topology .
108-415: In the simplest application, the case of a triangle on a plane , the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss , who developed a version but never published it, and Pierre Ossian Bonnet , who published a special case in 1848. Suppose M
216-415: A rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20. Euclid refers to a pair of lines, or
324-473: A torus has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary Riemannian metric from its embedding in R , then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on
432-402: A 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar . Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other. Because of Euclidean geometry's fundamental status in mathematics, it
540-460: A choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape , such as [REDACTED] , that moves continuously along such a loop is changed into its own mirror image [REDACTED] . A Möbius strip
648-425: A choice of clockwise and counter-clockwise. These two situations share the common feature that they are described in terms of top-dimensional behavior near p but not at p . For the general case, let M be a topological n -manifold. A local orientation of M around a point p is a choice of generator of the group To see the geometric significance of this group, choose a chart around p . In that chart there
756-577: A colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers. Apollonius of Perga ( c. 240 BCE – c. 190 BCE )
864-460: A connected double covering; this cover is orientable by construction. In the former case, one can simply take two copies of M {\displaystyle M} , each of which corresponds to a different orientation. A real vector bundle , which a priori has a GL(n) structure group , is called orientable when the structure group may be reduced to G L + ( n ) {\displaystyle GL^{+}(n)} ,
972-453: A decision of whether, in the given chart, a sphere around p is positive or negative. A reflection of R through the origin acts by negation on H n − 1 ( S n − 1 ; Z ) {\displaystyle H_{n-1}\left(S^{n-1};\mathbf {Z} \right)} , so the geometric significance of the choice of generator is that it distinguishes charts from their reflections. On
1080-460: A function, Descartes' theorem is Gauss–Bonnet where the curvature is a discrete measure , and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem. There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let M be a finite 2-dimensional pseudo-manifold . Let χ ( v ) denote the number of triangles containing
1188-531: A generator of the infinite cyclic group H n ( M ; Z ) {\displaystyle H_{n}(M;\mathbf {Z} )} and taking the oriented charts to be those for which α pushes forward to the fixed generator. Conversely, an oriented atlas determines such a generator as compatible local orientations can be glued together to give a generator for the homology group H n ( M ; Z ) {\displaystyle H_{n}(M;\mathbf {Z} )} . A manifold M
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#17328024162571296-515: A loop going around the opposite way. This turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip . Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability. For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an orientation , and the surface
1404-404: A midpoint). Orientable manifold In mathematics , orientability is a property of some topological spaces such as real vector spaces , Euclidean spaces , surfaces , and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and
1512-433: A near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough. In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as R above)
1620-425: A pair of characters : the space orientation character σ + and the time orientation character σ − , Their product σ = σ + σ − is the determinant, which gives the orientation character. A space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated bundle where O( M ) is the bundle of pseudo-orthogonal frames. Similarly,
1728-427: A pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term " congruent " refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example,
1836-404: A ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. The pons asinorum ( bridge of asses ) states that in isosceles triangles
1944-437: A small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems. The Elements begins with plane geometry , still taught in secondary school (high school) as
2052-406: A sphere around p , and this sphere determines a generator of H n ( M , M ∖ { p } ; Z ) {\displaystyle H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)} . Moreover, any other chart around p is related to the first chart by an orientation preserving transition function, and this implies that the two charts yield
2160-485: A sphere of radius R . Its Euler characteristic is 1. On the left hand side of the theorem, we have K = 1 / R 2 {\displaystyle K=1/R^{2}} and k g = 0 {\displaystyle k_{g}=0} , because the boundary is the equator and the equator is a geodesic of the sphere. Then ∫ M K d A = 2 π {\displaystyle \int _{M}KdA=2\pi } . On
2268-444: A standard volume form given by dx ∧ ⋯ ∧ dx . Given a volume form on M , the collection of all charts U → R for which the standard volume form pulls back to a positive multiple of ω is an oriented atlas. The existence of a volume form is therefore equivalent to orientability of the manifold. Volume forms and tangent vectors can be combined to give yet another description of orientability. If X 1 , …, X n
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#17328024162572376-445: A statement such as "Find the greatest common measure of ..." Euclid often used proof by contradiction . Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C. Angles whose sum
2484-447: A steep bridge that only a sure-footed donkey could cross. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of
2592-421: A topological manifold, a transition function is orientation preserving if, at each point p in its domain, it fixes the generators of H n ( M , M ∖ { p } ; Z ) {\displaystyle H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)} . From here, the relevant definitions are the same as in the differentiable case. An oriented atlas
2700-408: A topology and the projection sending ( x , o ) {\displaystyle (x,o)} to x {\displaystyle x} is then a 2-to-1 covering map. This covering space is called the orientable double cover , as it is orientable. M ∗ {\displaystyle M^{*}} is connected if and only if M {\displaystyle M}
2808-431: A topology, and this topology makes it into a manifold. More precisely, let O be the set of all local orientations of M . To topologize O we will specify a subbase for its topology. Let U be an open subset of M chosen such that H n ( M , M ∖ U ; Z ) {\displaystyle H_{n}(M,M\setminus U;\mathbf {Z} )} is isomorphic to Z . Assume that α
2916-413: A triangulation, and in general for n > 4 some n -manifolds have triangulations that are inequivalent. If H 1 ( S ) denotes the first homology group of a closed surface S , then S is orientable if and only if H 1 ( S ) has a trivial torsion subgroup . More precisely, if S is orientable then H 1 ( S ) is a free abelian group , and if not then H 1 ( S ) = F + Z /2 Z where F
3024-453: Is constructive . Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge . In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory , which often assert
3132-520: Is a compact two-dimensional Riemannian manifold with boundary ∂ M . Let K be the Gaussian curvature of M , and let k g be the geodesic curvature of ∂ M . Then where dA is the element of area of the surface, and ds is the line element along the boundary of M . Here, χ ( M ) is the Euler characteristic of M . If the boundary ∂ M is piecewise smooth , then we interpret
3240-459: Is a fiber bundle with structure group GL( n , R ) . That is, the transition functions of the manifold induce transition functions on the tangent bundle which are fiberwise linear transformations. If the structure group can be reduced to the group GL ( n , R ) of positive determinant matrices, or equivalently if there exists an atlas whose transition functions determine an orientation preserving linear transformation on each tangent space, then
3348-411: Is a basis of tangent vectors at a point p , then the basis is said to be right-handed if ω( X 1 , …, X n ) > 0 . A transition function is orientation preserving if and only if it sends right-handed bases to right-handed bases. The existence of a volume form implies a reduction of the structure group of the tangent bundle or the frame bundle to GL ( n , R ) . As before, this implies
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3456-492: Is a generator of this group. For each p in U , there is a pushforward function H n ( M , M ∖ U ; Z ) → H n ( M , M ∖ { p } ; Z ) {\displaystyle H_{n}(M,M\setminus U;\mathbf {Z} )\to H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)} . The codomain of this group has two generators, and α maps to one of them. The topology on O
3564-496: Is a neighborhood of p which is an open ball B around the origin O . By the excision theorem , H n ( M , M ∖ { p } ; Z ) {\displaystyle H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)} is isomorphic to H n ( B , B ∖ { O } ; Z ) {\displaystyle H_{n}\left(B,B\setminus \{O\};\mathbf {Z} \right)} . The ball B
3672-403: Is a right angle are called complementary . Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary . Supplementary angles are formed when a ray shares the same vertex and
3780-404: Is an atlas for which all transition functions are orientation preserving. M is orientable if it admits an oriented atlas. When n > 0 , an orientation of M is a maximal oriented atlas. (When n = 0 , an orientation of M is a function M → {±1} .) Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle , so it
3888-411: Is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory , whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms . A generalization of
3996-475: Is called oriented . For surfaces embedded in Euclidean space, an orientation is specified by the choice of a continuously varying surface normal n at every point. If such a normal exists at all, then there are always two ways to select it: n or − n . More generally, an orientable surface admits exactly two orientations, and the distinction between an orient ed surface and an orient able surface
4104-583: Is contractible, so its homology groups vanish except in degree zero, and the space B \ O is an ( n − 1) -sphere, so its homology groups vanish except in degrees n − 1 and 0 . A computation with the long exact sequence in relative homology shows that the above homology group is isomorphic to H n − 1 ( S n − 1 ; Z ) ≅ Z {\displaystyle H_{n-1}\left(S^{n-1};\mathbf {Z} \right)\cong \mathbf {Z} } . A choice of generator therefore corresponds to
4212-418: Is defined so that is open. There is a canonical map π : O → M that sends a local orientation at p to p . It is clear that every point of M has precisely two preimages under π . In fact, π is even a local homeomorphism, because the preimages of the open sets U mentioned above are homeomorphic to the disjoint union of two copies of U . If M is orientable, then M itself
4320-404: Is fixed. Let U → R + be a chart at a boundary point of M which, when restricted to the interior of M , is in the chosen oriented atlas. The restriction of this chart to ∂ M is a chart of ∂ M . Such charts form an oriented atlas for ∂ M . When M is smooth, at each point p of ∂ M , the restriction of the tangent bundle of M to ∂ M is isomorphic to T p ∂ M ⊕ R , where
4428-461: Is free abelian, and the Z /2 Z factor is generated by the middle curve in a Möbius band embedded in S . Let M be a connected topological n - manifold . There are several possible definitions of what it means for M to be orientable. Some of these definitions require that M has extra structure, like being differentiable. Occasionally, n = 0 must be made into a special case. When more than one of these definitions applies to M , then M
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4536-438: Is impractical to give more than a representative sampling of applications here. As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is surveying . In addition it has been used in classical mechanics and the cognitive and computational approaches to visual perception of objects . Certain practical results from Euclidean geometry (such as
4644-554: Is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. There are 13 books in the Elements : Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and
4752-512: Is mainly known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry , an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian ( x , y ) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In
4860-576: Is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field ). Euclidean geometry is an example of synthetic geometry , in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to analytic geometry , introduced almost 2,000 years later by René Descartes , which uses coordinates to express geometric properties by means of algebraic formulas . The Elements
4968-399: Is not orientable. Another way to construct this cover is to divide the loops based at a basepoint into either orientation-preserving or orientation-reversing loops. The orientation preserving loops generate a subgroup of the fundamental group which is either the whole group or of index two. In the latter case (which means there is an orientation-reversing path), the subgroup corresponds to
5076-404: Is one for which all transition functions are orientation preserving, M is orientable if it admits an oriented atlas, and when n > 0 , an orientation of M is a maximal oriented atlas. Intuitively, an orientation of M ought to define a unique local orientation of M at each point. This is made precise by noting that any chart in the oriented atlas around p can be used to determine
5184-421: Is one of these open sets, so O is the disjoint union of two copies of M . If M is non-orientable, however, then O is connected and orientable. The manifold O is called the orientation double cover . If M is a manifold with boundary, then an orientation of M is defined to be an orientation of its interior. Such an orientation induces an orientation of ∂ M . Indeed, suppose that an orientation of M
5292-616: Is orientable if and only if the first Stiefel–Whitney class w 1 ( M ) ∈ H 1 ( M ; Z / 2 ) {\displaystyle w_{1}(M)\in H^{1}(M;\mathbf {Z} /2)} vanishes. In particular, if the first cohomology group with Z /2 coefficients is zero, then the manifold is orientable. Moreover, if M is orientable and w 1 vanishes, then H 0 ( M ; Z / 2 ) {\displaystyle H^{0}(M;\mathbf {Z} /2)} parametrizes
5400-489: Is orientable under one definition if and only if it is orientable under the others. The most intuitive definitions require that M be a differentiable manifold. This means that the transition functions in the atlas of M are C -functions. Such a function admits a Jacobian determinant . When the Jacobian determinant is positive, the transition function is said to be orientation preserving . An oriented atlas on M
5508-466: Is orientable. For example, a torus embedded in can be one-sided, and a Klein bottle in the same space can be two-sided; here K 2 {\displaystyle K^{2}} refers to the Klein bottle. Any surface has a triangulation : a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. Each triangle is oriented by choosing a direction around
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#17328024162575616-444: Is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite. In modern terminology, angles would normally be measured in degrees or radians . Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing
5724-485: Is proved that there are infinitely many prime numbers. Books XI–XIII concern solid geometry . A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system , in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry , these axioms were considered to be obviously true in
5832-713: Is subtle and frequently blurred. An orientable surface is an abstract surface that admits an orientation, while an oriented surface is a surface that is abstractly orientable, and has the additional datum of a choice of one of the two possible orientations. Most surfaces encountered in the physical world are orientable. Spheres , planes , and tori are orientable, for example. But Möbius strips , real projective planes , and Klein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side. The real projective plane and Klein bottle cannot be embedded in R , only immersed with nice intersections. Note that locally an embedded surface always has two sides, so
5940-499: Is the determination of packing arrangements , such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction . Geometry is used extensively in architecture . Geometry can be used to design origami . Some classical construction problems of geometry are impossible using compass and straightedge , but can be solved using origami . Archimedes ( c. 287 BCE – c. 212 BCE ),
6048-501: The 2 n -dimensional generalization of GB (also see Chern–Weil homomorphism ). The Riemann–Roch theorem can also be seen as a generalization of GB to complex manifolds . A far-reaching generalization that includes all the abovementioned theorems is the Atiyah–Singer index theorem . A generalization to 2-manifolds that need not be compact is Cohn-Vossen's inequality . In Greg Egan 's novel Diaspora , two characters discuss
6156-423: The Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) Books V and VII–X deal with number theory , with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It
6264-550: The right angle as his basic unit, so that, for example, a 45- degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from distances. For example,
6372-680: The Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation defining the distance between two points P = ( p x , p y ) and Q = ( q x , q y ) is then known as the Euclidean metric , and other metrics define non-Euclidean geometries . In terms of analytic geometry,
6480-412: The amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°. The area of a hyperbolic triangle , conversely is proportional to its defect , as established by Johann Heinrich Lambert . Descartes' theorem on total angular defect of a polyhedron is the polyhedral analog: it states that the sum of the defect at all
6588-406: The angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another . Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to
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#17328024162576696-426: The angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle . The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite
6804-407: The area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder. Euclidean geometry has two fundamental types of measurements: angle and distance . The angle scale is absolute, and Euclid uses
6912-410: The border, which is a geodesic triangle. But we have three right-angle corners, so ∫ ∂ M k g d s = 3 π 2 {\displaystyle \int _{\partial M}k_{g}ds={\frac {3\pi }{2}}} . The theorem applies in particular to compact surfaces without boundary, in which case the integral can be omitted. It states that
7020-455: The case of the plane (where the Gaussian curvature is 0 and geodesics are straight lines), we recover the familiar formula for the sum of angles in an ordinary triangle. On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger than π . A number of earlier results in spherical geometry and hyperbolic geometry, discovered over
7128-563: The choices of orientations. This characterization of orientability extends to orientability of general vector bundles over M , not just the tangent bundle. Around each point of M there are two local orientations. Intuitively, there is a way to move from a local orientation at a point p to a local orientation at a nearby point p ′ : when the two points lie in the same coordinate chart U → R , that coordinate chart defines compatible local orientations at p and p ′ . The set of local orientations can therefore be given
7236-404: The cube and squaring the circle . In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation. Euler discussed a generalization of Euclidean geometry called affine geometry , which retains
7344-420: The dent. Compactness of the surface is of crucial importance. Consider for instance the open unit disc , a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work. It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2 π . As an application,
7452-500: The derivation of this theorem. The theorem can be used directly as a system to control sculpture - for example, in work by Edmund Harriss in the collection of the University of Arkansas Honors College . Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid , which he described in his textbook on geometry , Elements . Euclid's approach consists in assuming
7560-730: The existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider nonconstructive proofs just as sound as constructive ones, they are often considered less elegant , intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring
7668-446: The existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique. The Elements also include the following five "common notions": Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than
7776-522: The factor of R is described by the inward pointing normal vector. The orientation of T p ∂ M is defined by the condition that a basis of T p ∂ M is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis of T p M . A closely related notion uses the idea of covering space . For a connected manifold M {\displaystyle M} take M ∗ {\displaystyle M^{*}} ,
7884-423: The fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have
7992-401: The first axiomatic system and the first examples of mathematical proofs . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with
8100-399: The formula above with 2 and 4, respectively. More specifically, if M is a closed 2-dimensional digital manifold , the genus turns out where M i indicates the number of surface-points each of which has i adjacent points on the surface. This is the simplest formula of Gauss–Bonnet theorem in three-dimensional digital space. The Chern theorem (after Shiing-Shen Chern 1945) is
8208-410: The group of matrices with positive determinant . For the tangent bundle , this reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a smooth real manifold : a smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right,
8316-416: The integral ∫ ∂ M k g ds as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary. Many standard proofs use the theorem of turning tangents, which states roughly that the winding number of a Jordan curve is exactly ±1. Suppose M is the northern hemisphere cut out from
8424-446: The manifold M is orientable. Conversely, M is orientable if and only if the structure group of the tangent bundle can be reduced in this way. Similar observations can be made for the frame bundle. Another way to define orientations on a differentiable manifold is through volume forms . A volume form is a nowhere vanishing section ω of ⋀ T M , the top exterior power of the cotangent bundle of M . For example, R has
8532-429: The manifold because an orientation of the manifold is an atlas, and it makes no sense to say that a transition function preserves or does not preserve an atlas of which it is a member. This question can be resolved by defining local orientations. On a one-dimensional manifold, a local orientation around a point p corresponds to a choice of left and right near that point. On a two-dimensional manifold, it corresponds to
8640-439: The manner of Euclid Book III, Prop. 31. In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, A ∝ L 2 {\displaystyle A\propto L^{2}} , and the volume of a solid to the cube, V ∝ L 3 {\displaystyle V\propto L^{3}} . Euclid proved these results in various special cases such as
8748-433: The notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle ) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values. A surface S in the Euclidean space R is orientable if a chiral two-dimensional figure (for example, [REDACTED] ) cannot be moved around
8856-430: The number of special cases is reduced. Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example,
8964-506: The orientability of M . Conversely, if M is orientable, then local volume forms can be patched together to create a global volume form, orientability being necessary to ensure that the global form is nowhere vanishing. At the heart of all the above definitions of orientability of a differentiable manifold is the notion of an orientation preserving transition function. This raises the question of what exactly such transition functions are preserving. They cannot be preserving an orientation of
9072-462: The other hand, suppose we flatten the hemisphere to make it into a disk. This transformation is a homeomorphism, so the Euler characteristic is still 1. However, on the left hand side of the theorem we now have K = 0 {\displaystyle K=0} and k g = 1 / R {\displaystyle k_{g}=1/R} , because a circumference is not a geodesic of
9180-466: The others, as evidenced by the organization of the Elements : his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated which are logically equivalent to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states: The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. Euclidean Geometry
9288-462: The others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from
9396-458: The perimeter of the triangle, associating a direction to each edge of the triangle. If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface. Such a choice is only possible if the surface is orientable, and in this case there are exactly two different orientations. If the figure [REDACTED] can be consistently positioned at all points of
9504-410: The physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. Near the beginning of the first book of the Elements , Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): Although Euclid explicitly only asserts
9612-633: The plane. Then ∫ ∂ M k g d s = 2 π {\displaystyle \int _{\partial M}k_{g}ds=2\pi } . Finally, take a sphere octant, also homeomorphic to the previous cases. Then ∫ M K d A = 1 R 2 4 π R 2 8 = π 2 {\displaystyle \int _{M}KdA={\frac {1}{R^{2}}}{\frac {4\pi R^{2}}{8}}={\frac {\pi }{2}}} . Now k g = 0 {\displaystyle k_{g}=0} almost everywhere along
9720-409: The possible exception of the parallel postulate ) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein 's theory of general relativity is that physical space itself
9828-424: The preceding centuries, were subsumed as special cases of Gauss–Bonnet. In spherical trigonometry and hyperbolic trigonometry , the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°. The area of a spherical triangle is proportional to its excess, by Girard's theorem –
9936-437: The problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling
10044-533: The restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2 x + 1 (a line), or x + y = 7 (a circle). Also in the 17th century, Girard Desargues , motivated by the theory of perspective , introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometry , but it can also be used to produce proofs in ordinary Euclidean geometry in which
10152-413: The right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem , named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after
10260-428: The right-angle property of the 3-4-5 triangle) were used long before they were proved formally. The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as Gunter's chain , and angles using graduated circles and, later, the theodolite . An application of Euclidean solid geometry
10368-479: The same generator, whence the generator is unique. Purely homological definitions are also possible. Assuming that M is closed and connected, M is orientable if and only if the n th homology group H n ( M ; Z ) {\displaystyle H_{n}(M;\mathbf {Z} )} is isomorphic to the integers Z . An orientation of M is a choice of generator α of this group. This generator determines an oriented atlas by fixing
10476-439: The same spacetime point, and then meet again at another point, they remain right-handed with respect to one another. If a spacetime is time-orientable then the two observers will always agree on the direction of time at both points of their meeting. In fact, a spacetime is time-orientable if and only if any two observers can agree which of the two meetings preceded the other. Formally, the pseudo-orthogonal group O( p , q ) has
10584-430: The set of pairs ( x , o ) {\displaystyle (x,o)} where x {\displaystyle x} is a point of M {\displaystyle M} and o {\displaystyle o} is an orientation at x {\displaystyle x} ; here we assume M {\displaystyle M} is either smooth so we can choose an orientation on
10692-438: The surface M , its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that the total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4 π (the Euler characteristic of a sphere being 2), no matter how big or deep
10800-440: The surface T formed by the inside of that triangle and the piecewise boundary of the triangle. The geodesic curvature the bordering geodesics is 0, and the Euler characteristic of T being 1. Hence the sum of the turning angles of the geodesic triangle is equal to 2 π minus the total curvature within the triangle. Since the turning angle at a corner is equal to π minus the interior angle, we can rephrase this as follows: In
10908-451: The surface and back to where it started so that it looks like its own mirror image ( [REDACTED] ). Otherwise the surface is non-orientable . An abstract surface (i.e., a two-dimensional manifold ) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. That is to say that a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to
11016-415: The surface without turning into its mirror image, then this will induce an orientation in the above sense on each of the triangles of the triangulation by selecting the direction of each of the triangles based on the order red-green-blue of colors of any of the figures in the interior of the triangle. This approach generalizes to any n -manifold having a triangulation. However, some 4-manifolds do not have
11124-412: The tangent bundle is always orientable, even over nonorientable manifolds. In Lorentzian geometry , there are two kinds of orientability: space orientability and time orientability . These play a role in the causal structure of spacetime. In the context of general relativity , a spacetime manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at
11232-395: The tangent space at a point or we use singular homology to define orientation. Then for every open, oriented subset of M {\displaystyle M} we consider the corresponding set of pairs and define that to be an open set of M ∗ {\displaystyle M^{*}} . This gives M ∗ {\displaystyle M^{*}}
11340-441: The torus is flat and has constant curvature 0, again resulting in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature. Sometimes the Gauss–Bonnet formula is stated as where T is a geodesic triangle . Here we define a "triangle" on M to be a simply connected region whose boundary consists of three geodesics . We can then apply GB to
11448-445: The total Gaussian curvature of such a closed surface is equal to 2 π times the Euler characteristic of the surface. Note that for orientable compact surfaces without boundary, the Euler characteristic equals 2 − 2 g , where g is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and g counts the number of handles. If one bends and deforms
11556-540: The vertex v . Then where the first sum ranges over the vertices in the interior of M , the second sum is over the boundary vertices, and χ ( M ) is the Euler characteristic of M . Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of n vertices, we must replace 3 and 6 in the formula above with n / n − 2 and 2 n / n − 2 , respectively. For example, for quadrilaterals we must replace 3 and 6 in
11664-413: The vertices of a polyhedron which is homeomorphic to the sphere is 4 π . More generally, if the polyhedron has Euler characteristic χ = 2 − 2 g (where g is the genus, meaning "number of holes"), then the sum of the defect is 2 πχ . This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices). Thinking of curvature as a measure , rather than as
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