Misplaced Pages

#382617

74-448: It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces ) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry . The Riemann–Roch theorem and

148-474: 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

222-727: A Euclidean space. Weil met Chern in Princeton after Chern arrived in August 1943. He told Chern that he believed there should be an intrinsic proof, which Chern was able to obtain within two weeks. The result is Chern's classic paper "A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds" published in the Annals of Mathematics the next year. The earlier work of Allendoerfer, Fenchel, Allendoerfer and Weil were cited by Chern in this paper. The work of Allendoerfer and Weil

296-576: A Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz–Killing curvature (the average of the Gauss–Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold). Their result would be valid for the general case if

370-568: A compact, even-dimensional hypersurface M {\displaystyle M} in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} we get where d V {\displaystyle dV} is the volume element of the hypersurface, K {\displaystyle K} is the Jacobian determinant of the Gauss map , and γ n {\displaystyle \gamma _{n}}

444-526: A final 1 on the diagonal. The {±1} component is represented by block-diagonal matrices with 2-by-2 blocks either with the last component ±1 chosen to make the determinant 1 . The Weyl group of SO(2 n ) is the subgroup H n − 1 ⋊ S n < { ± 1 } n ⋊ S n {\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}} of that of SO(2 n + 1) , where H n −1 < {±1}

518-431: A fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group , by analogy with the general linear group . Equivalently, it is the group of n × n orthogonal matrices , where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose ). The orthogonal group

592-405: 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

666-450: A non-degenerate symmetric bilinear form or quadratic form on a vector space over a field , the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product , or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups , since

740-564: A normal elementary abelian 2-subgroup and a symmetric group , where the nontrivial element of each {±1} factor of {±1} acts on the corresponding circle factor of T × {1 } by inversion , and the symmetric group S n acts on both {±1} and T × {1 } by permuting factors. The elements of the Weyl group are represented by matrices in O(2 n ) × {±1} . The S n factor is represented by block permutation matrices with 2-by-2 blocks, and

814-403: A rotation by π and a pair of eigenvalues +1 can be identified with a rotation by 0 . The special case of n = 3 is known as Euler's rotation theorem , which asserts that every (non-identity) element of SO(3) is a rotation about a unique axis–angle pair. Reflections are the elements of O( n ) whose canonical form is where I is the ( n − 1) × ( n − 1) identity matrix, and

SECTION 10

#1732793525383

888-597: A skew-symmetric 2 n × 2 n matrix whose entries are 2-forms, so it is a matrix over the commutative ring ⋀ even T ∗ M {\textstyle {\bigwedge }^{\text{even}}\,T^{*}M} . Hence the Pfaffian is a 2 n -form. It is also an invariant polynomial . However, Chern's theorem in general is that for any closed C ∞ {\displaystyle C^{\infty }} orientable n -dimensional M {\displaystyle M} , where

962-578: A special instance in the theory of characteristic classes . The Chern integrand is the Euler class . Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when changing the Riemannian metric , one stays in the same cohomology class . That means that the integral of the Euler class remains constant as the metric is varied and is thus a global invariant of

1036-469: A subgroup called the special orthogonal group , denoted SO( n ) , consisting of all direct isometries of O( n ) , which are those that preserve the orientation of the space. SO( n ) is a normal subgroup of O( n ) , as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group {−1, +1} . This implies that the orthogonal group is an internal semidirect product of SO( n ) and any subgroup formed with

1110-414: 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 is a compact two-dimensional Riemannian manifold with boundary ∂ M . Let K be

1184-536: Is a compact orientable 2 n -dimensional Riemannian manifold without boundary , and Ω {\displaystyle \Omega } is the associated curvature form of the Levi-Civita connection . In fact, the statement holds with Ω {\displaystyle \Omega } the curvature form of any metric connection on the tangent bundle, as well as for other vector bundles over M {\displaystyle M} . Since

1258-570: Is a manifold with boundary . A far-reaching generalization of the Gauss–Bonnet theorem is the Atiyah–Singer Index Theorem . Let D {\displaystyle D} be a weakly elliptic differential operator between vector bundles. That means that the principal symbol is an isomorphism . Strong ellipticity would furthermore require the symbol to be positive-definite . Let D ∗ {\displaystyle D^{*}} be its adjoint operator . Then

1332-477: Is a basis on which the torus consists of the block-diagonal matrices of the form where each R j belongs to SO(2) . In O(2 n + 1) and SO(2 n + 1) , the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal. The Weyl group of SO(2 n + 1) is the semidirect product { ± 1 } n ⋊ S n {\displaystyle \{\pm 1\}^{n}\rtimes S_{n}} of

1406-487: Is a natural group homomorphism p from E( n ) to O( n ) , which is defined by where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see Affine space § Subtraction and Weyl's axioms ). The kernel of p

1480-438: Is an algebraic group and a Lie group . It is compact . The orthogonal group in dimension n has two connected components . The one that contains the identity element is a normal subgroup , called the special orthogonal group , and denoted SO( n ) . It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group , generalizing the fact that in dimensions 2 and 3, its elements are

1554-562: Is called an orthogonal matrix over F . The n × n orthogonal matrices form a subgroup, denoted O( n , F ) , of the general linear group GL( n , F ) ; that is O ⁡ ( n , F ) = { Q ∈ GL ⁡ ( n , F ) ∣ Q T Q = Q Q T = I } . {\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.} More generally, given

SECTION 20

#1732793525383

1628-413: Is the kernel of the product homomorphism {±1} → {±1} given by ( ε 1 , … , ε n ) ↦ ε 1 ⋯ ε n {\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}} ; that is, H n −1 < {±1}

1702-470: Is the scalar curvature . This is particularly important in general relativity , where spacetime is viewed as a 4-dimensional manifold. In terms of the orthogonal Ricci decomposition of the Riemann curvature tensor, this formula can also be written as where W {\displaystyle W} is the Weyl tensor and Z {\displaystyle Z} is the traceless Ricci tensor. For

1776-464: Is the surface area of the unit n-sphere . The Gauss–Bonnet theorem is a special case when M {\displaystyle M} is a 2-dimensional manifold. It arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss–Bonnet integrand. As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when M {\displaystyle M}

1850-457: Is the symmetry group of the ( n − 1) -sphere (for n = 3 , this is just the sphere ) and all objects with spherical symmetry, if the origin is chosen at the center. The symmetry group of a circle is O(2) . The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the circle group , also known as U (1) , the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends

1924-505: Is the subgroup of the general linear group GL( n , R ) , consisting of all endomorphisms that preserve the Euclidean norm ; that is, endomorphisms g such that ‖ g ( x ) ‖ = ‖ x ‖ . {\displaystyle \|g(x)\|=\|x\|.} Let E( n ) be the group of the Euclidean isometries of a Euclidean space S of dimension n . This group does not depend on

1998-734: Is the subgroup with an even number of minus signs. The Weyl group of SO(2 n ) is represented in SO(2 n ) by the preimages under the standard injection SO(2 n ) → SO(2 n + 1) of the representatives for the Weyl group of SO(2 n + 1) . Those matrices with an odd number of [ 0 1 1 0 ] {\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}} blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2 n ) . The low-dimensional (real) orthogonal groups are familiar spaces : In terms of algebraic topology , for n > 2

2072-464: Is the vector space of the translations. So, the translations form a normal subgroup of E( n ) , the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O( n ) . Moreover, the Euclidean group is a semidirect product of O( n ) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to

2146-501: 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

2220-636: The Atiyah–Singer index theorem are other generalizations of the Gauss–Bonnet theorem. One useful form of the Chern theorem is that where χ ( M ) {\displaystyle \chi (M)} denotes the Euler characteristic of M {\displaystyle M} . The Euler class is defined as where we have the Pfaffian Pf ⁡ ( Ω ) {\displaystyle \operatorname {Pf} (\Omega )} . Here M {\displaystyle M}

2294-520: The Euler characteristic vanishes for odd dimensions. There is some research being done on 'twisting' the index theorem in K-theory to give non-trivial results for odd dimensions. There is also a version of Chern's formula for orbifolds . Shiing-Shen Chern published his proof of the theorem in 1944 while at the Institute for Advanced Study . This was historically the first time that the formula

Chern–Gauss–Bonnet theorem - Misplaced Pages Continue

2368-474: 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 the integral ∫ ∂ M k g ds as the sum of the corresponding integrals along

2442-461: The Nash embedding theorem can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian manifolds in 1956. In 1943 Allendoerfer and Weil published their proof for the general case, in which they first used an approximation theorem of H. Whitney to reduce the case to analytic Riemannian manifolds, then they embedded "small" neighborhoods of

2516-483: The analytical index is defined as By ellipticity this is always finite. The index theorem says that this is constant as the elliptic operator is varied smoothly. It is equal to a topological index , which can be expressed in terms of characteristic classes like the Euler class . The Chern–Gauss–Bonnet theorem is derived by considering the Dirac operator The Chern formula is only defined for even dimensions because

2590-401: The cyclic group C k of k -fold rotations , for every positive integer k . All these groups are normal subgroups of O(2) and SO(2) . For any element of O( n ) there is an orthogonal basis, where its matrix has the form where there may be any number, including zero, of ±1's; and where the matrices R 1 , ..., R k are 2-by-2 rotation matrices, that is matrices of

2664-491: The fundamental group of SO( n , R ) is cyclic of order 2 , and the spin group Spin( n ) is its universal cover . For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique connected 2-fold cover ). Generally, the homotopy groups π k ( O ) of the real orthogonal group are related to homotopy groups of spheres , and thus are in general hard to compute. However, one can compute

2738-400: The identity component , that is, the connected component containing the identity matrix . The orthogonal group O( n ) can be identified with the group of the matrices A such that A A = I . Since both members of this equation are symmetric matrices , this provides n ( n + 1) / 2 equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by

2812-421: The above canonical form and the case of dimension two. The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two. The reflection through the origin (the map v ↦ − v ) is an example of an element of O( n ) that is not a product of fewer than n reflections. The orthogonal group O( n )

2886-505: The above pairing (,) denotes the cap product with the Euler class of the tangent bundle T M {\displaystyle TM} . In 1944, the general theorem was first proved by S. S. Chern in a classic paper published by the Princeton University math department. In 2013, a proof of the theorem via supersymmetric Euclidean field theories was also found. The Chern–Gauss–Bonnet theorem can be seen as

2960-412: 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

3034-410: 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

Chern–Gauss–Bonnet theorem - Misplaced Pages Continue

3108-455: 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

3182-433: The choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic . The stabilizer subgroup of a point x ∈ S is the subgroup of the elements g ∈ E( n ) such that g ( x ) = x . This stabilizer is (or, more exactly, is isomorphic to) O( n ) , since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space. There

3256-568: The complex number exp( φ i ) = cos( φ ) + i sin( φ ) of absolute value   1 to the special orthogonal matrix In higher dimension, O( n ) has a more complicated structure (in particular, it is no longer commutative). The topological structures of the n -sphere and O( n ) are strongly correlated, and this correlation is widely used for studying both topological spaces . The groups O( n ) and SO( n ) are real compact Lie groups of dimension n ( n − 1) / 2 . The group O( n ) has two connected components , with SO( n ) being

3330-446: The condition of preserving a form can be expressed as an equality of matrices. The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space E of dimension n , the elements of the orthogonal group O( n ) are, up to a uniform scaling ( homothecy ), the linear maps from E to E that map orthogonal vectors to orthogonal vectors. The orthogonal O( n )

3404-420: 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,

3478-494: 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 . Special orthogonal group In mathematics , the orthogonal group in dimension n , denoted O( n ) , is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve

3552-442: The determinant (that is det( A ) = 1 or det( A ) = −1 ). Both are nonsingular algebraic varieties of the same dimension n ( n − 1) / 2 . The component with det( A ) = 1 is SO( n ) . A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to T for some k , where T = SO(2) is the standard one-dimensional torus. In O(2 n ) and SO(2 n ) , for every maximal torus, there

3626-418: The dimension is 2 n , we have that Ω {\displaystyle \Omega } is an s o ( 2 n ) {\displaystyle {\mathfrak {s}}{\mathfrak {o}}(2n)} -valued 2-differential form on M {\displaystyle M} (see special orthogonal group ). So Ω {\displaystyle \Omega } can be regarded as

3700-407: The entries of any non-orthogonal matrix. This proves that O( n ) is an algebraic set . Moreover, it can be proved that its dimension is which implies that O( n ) is a complete intersection . This implies that all its irreducible components have the same dimension, and that it has no embedded component . In fact, O( n ) has two irreducible components, that are distinguished by the sign of

3774-400: The form with a + b = 1 . This results from the spectral theorem by regrouping eigenvalues that are complex conjugate , and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1 . The element belongs to SO( n ) if and only if there are an even number of −1 on the diagonal. A pair of eigenvalues −1 can be identified with

SECTION 50

#1732793525383

3848-400: 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

3922-450: The homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions: Since the inclusions are all closed, hence cofibrations , this can also be interpreted as a union. On the other hand, S is a homogeneous space for O( n + 1) , and one has the following fiber bundle : which can be understood as "The orthogonal group O( n + 1) acts transitively on

3996-419: The identity and a reflection . The group with two elements {± I } (where I is the identity matrix) is a normal subgroup and even a characteristic subgroup of O( n ) , and, if n is even, also of SO( n ) . If n is odd, O( n ) is the internal direct product of SO( n ) and {± I } . The group SO(2) is abelian (whereas SO( n ) is not abelian when n > 2 ). Its finite subgroups are

4070-502: The manifold isometrically into a Euclidean space with the help of the Cartan–Janet local embedding theorem, so that they can patch these embedded neighborhoods together and apply the above theorem of Allendoerfer and Fenchel to establish the global result. This is, of course, unsatisfactory for the reason that the theorem only involves intrinsic invariants of the manifold, then the validity of the theorem should not rely on its embedding into

4144-410: 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

4218-633: 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

4292-424: 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 –

4366-408: 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 a sphere of radius R . Its Euler characteristic is 1. On the left hand side of

4440-519: The smooth structure. The theorem has also found numerous applications in physics , including: In dimension 2 n = 4 {\displaystyle 2n=4} , for a compact oriented manifold, we get where Riem {\displaystyle {\text{Riem}}} is the full Riemann curvature tensor , Ric {\displaystyle {\text{Ric}}} is the Ricci curvature tensor , and R {\displaystyle R}

4514-443: The stable space equal the lower homotopy groups of the unstable spaces. From Bott periodicity we obtain Ω O ≃ O , therefore the homotopy groups of O are 8-fold periodic, meaning π k + 8 ( O ) = π k ( O ) , and so one need list only the first 8 homotopy groups: Via the clutching construction , homotopy groups of the stable space O are identified with stable vector bundles on spheres ( up to isomorphism ), with

SECTION 60

#1732793525383

4588-425: The study of O( n ) . By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices , which are the matrices such that It follows from this equation that the square of the determinant of Q equals 1 , and thus the determinant of Q is either 1 or −1 . The orthogonal matrices with determinant 1 form

4662-438: 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

4736-441: 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

4810-405: 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

4884-441: 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

4958-445: 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

5032-406: The unit sphere S , and the stabilizer of a point (thought of as a unit vector ) is the orthogonal group of the perpendicular complement , which is an orthogonal group one dimension lower." Thus the natural inclusion O( n ) → O( n + 1) is ( n − 1) -connected , so the homotopy groups stabilize, and π k (O( n + 1)) = π k (O( n )) for n > k + 1 : thus the homotopy groups of

5106-528: The usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2) , SO(3) and SO(4) . The other component consists of all orthogonal matrices of determinant −1 . This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component. By extension, for any field F , an n × n matrix with entries in F such that its inverse equals its transpose

5180-540: 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

5254-413: 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

5328-512: The zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane . In dimension two, every rotation can be decomposed into a product of two reflections . More precisely, a rotation of angle θ is the product of two reflections whose axes form an angle of θ / 2 . A product of up to n elementary reflections always suffices to generate any element of O( n ) . This results immediately from

5402-463: Was also cited by Chern in his second paper related to the same topic. Gauss%E2%80%93Bonnet theorem 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 . In the simplest application, the case of a triangle on a plane , the sum of its angles

5476-462: Was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a hypersurface (an (n-1)-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss–Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to

#382617