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112-406: Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry , proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in

224-456: A Fields medal , made new impacts in mathematics by using topological quantum field theory and string theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants . Riemannian geometry studies Riemannian manifolds , smooth manifolds with a Riemannian metric . This is

336-487: A directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor . Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry . This notion can also be defined locally , i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However,

448-476: A module over the ring of smooth functions, C ( M ), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any commutative ring . As a motivating example, consider the space Ω 1 ( M ) = T 1 0 ( M ) {\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)} of smooth covector fields ( 1-forms ), also

560-447: A vector bundle endomorphism (called an almost complex structure ) It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifold is called complex if N J = 0 {\displaystyle N_{J}=0} , where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called

672-505: A combinatorial and differential-geometric nature. Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma } for

784-440: A common pizza -eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in

896-595: A concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation . Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of

1008-540: A coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates. Following Schouten (1951) and McConnell (1957) , the concept of a tensor relies on a concept of a reference frame (or coordinate system ), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems. For example, coordinates belonging to

1120-496: A corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened. Mathematically, a sphere and a plane are not isometric , even locally. This fact is significant for cartography : it implies that no planar (flat) map of Earth can be perfect, even for

1232-412: A different type (although this is not usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C ) vector fields on M , X ( M ) := T 0 1 ( M ) {\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)} (see the section on notation above) as a single space —

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1344-491: A field concerned more generally with geometric structures on differentiable manifolds . A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over

1456-439: A fresh notion, the covariant derivative . This handles the formulation of variation of a tensor field along a vector field . The original absolute differential calculus notion, which was later called tensor calculus , led to the isolation of the geometric concept of connection . An extension of the tensor field idea incorporates an extra line bundle L on M . If W is the tensor product bundle of V with L , then W

1568-464: A function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously. A frequent example application of this general rule is showing that the Levi-Civita connection , which is a mapping of smooth vector fields ( X , Y ) ↦ ∇ X Y {\displaystyle (X,Y)\mapsto \nabla _{X}Y} taking

1680-506: A module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ω and a vector field X , we define Because of the pointwise nature of everything involved, the action of ω ~ {\displaystyle {\tilde {\omega }}} on X is a C ( M )-linear map, that is, for any p in M and smooth function f . Thus we can regard covector fields not just as sections of

1792-438: A more traditional explanation see the tensor density article. One feature of the bundle of densities (again assuming orientability) L is that L is well-defined for real number values of s ; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a half-density , the case where s = ⁠ 1 / 2 ⁠ . In general we can take sections of W ,

1904-461: A nondegenerate 2- form ω , called the symplectic form . A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0 . A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism . Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2,

2016-553: A pair of vector fields to a vector field, does not define a tensor field on M . This is because it is only R {\displaystyle \mathbb {R} } -linear in Y (in place of full C ( M )-linearity, it satisfies the Leibniz rule, ∇ X ( f Y ) = ( X f ) Y + f ∇ X Y {\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y} )). Nevertheless, it must be stressed that even though it

2128-504: A portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances. The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of

2240-557: A product of k copies of X ( M ) {\displaystyle {\mathfrak {X}}(M)} and l copies of Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} into C ( M ), it turns out that it arises from a tensor field on M if and only if it is multilinear over C ( M ). Namely C ∞ ( M ) {\displaystyle C^{\infty }(M)} -module of tensor fields of type ( k , l ) {\displaystyle (k,l)} over M

2352-480: A routine way – again independently of coordinates, as mentioned in the introduction. We therefore can give a definition of tensor field , namely as a section of some tensor bundle . (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of

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2464-433: A rudimentary measure of arclength of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s. Around this time there were only minimal overt applications of the theory of infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid 's Elements the notion of tangency of a line to a circle is discussed, and Archimedes applied

2576-513: A single bivector-valued one-form called the shape operator . Below are some examples of how differential geometry is applied to other fields of science and mathematics. Tensor field In mathematics and physics , a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold ) or of the physical space . Tensor fields are used in differential geometry , algebraic geometry , general relativity , in

2688-506: A student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation , and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in

2800-411: A subject begins at least as far back as classical antiquity . It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology , especially the study of manifolds . In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces , and eventually the development of

2912-515: A symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi 's and William Rowan Hamilton 's formulations of classical mechanics . By contrast with Riemannian geometry, where

3024-438: A system of functions v k {\displaystyle v^{k}} of the coordinates that, under such an affine transformation undergoes a transformation This is precisely the requirement needed to ensure that the quantity v k e k {\displaystyle v^{k}\mathbf {e} _{k}} is an invariant object that does not depend on the coordinate system chosen. More generally,

3136-563: A tensor A is defined on a vector fields set X(M) over a module M , we call A a tensor field on M . Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold , which is a topological space . Let M be a manifold , for instance the Euclidean plane R . Definition. A tensor field of type ( p , q )

3248-408: A tensor of valence ( p , q ) has p downstairs indices and q upstairs indices, with the transformation law being The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable , analytic , etc.). A covector field is a function v k {\displaystyle v_{k}} of the coordinates that transforms by

3360-474: A tensor section is not only a linear map on the vector space of sections, but a C ( M )-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus,

3472-581: A vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface. Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field g {\displaystyle g} , such that given any two vectors v , w {\displaystyle v,w} at point x {\displaystyle x} , their inner product

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3584-417: A well-known standard definition of metric and parallelism. In Riemannian geometry , the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and connections are related to various physical fields. From

3696-434: Is g x ( v , w ) {\displaystyle g_{x}(v,w)} . The field g {\displaystyle g} could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is Tissot's indicatrix . In general, we want to specify tensor fields in

3808-413: Is a vector space and the tensor bundle is a special kind of vector bundle . The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold M . For example, a vector space of one dimension depending on an angle could look like a Möbius strip or alternatively like a cylinder . Given a vector bundle V over M ,

3920-404: Is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the geometric nature of tensor quantities ; this kind of descent argument justifies abstractly the whole theory. The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it

4032-473: Is a bundle of vector spaces of just the same dimension as V . This allows one to define the concept of tensor density , a 'twisted' type of tensor field. A tensor density is the special case where L is the bundle of densities on a manifold , namely the determinant bundle of the cotangent bundle . (To be strictly accurate, one should also apply the absolute value to the transition functions – this makes little difference for an orientable manifold .) For

4144-401: Is a function F : T M → [ 0 , ∞ ) {\displaystyle F:\mathrm {T} M\to [0,\infty )} such that: Symplectic geometry is the study of symplectic manifolds . An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e.,

4256-475: Is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem .) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by

4368-402: Is a section where V is a vector bundle on M , V is its dual and ⊗ is the tensor product of vector bundles. Equivalently, it is a collection of elements T x ∈ V x ⊗ ( V x ) for all points x ∈ M , arranging into a smooth map T : M → V ⊗ ( V ) . Elements T x are called tensors . Often we take V = TM to be the tangent bundle of M . Intuitively,

4480-595: Is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the w th power, is called a tensor density with weight w . Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n -forms (where n is the dimension of the space), as opposed to taking their values in just R . Higher "weights" then just correspond to taking additional tensor products with this space in

4592-403: Is canonically isomorphic to C ∞ ( M ) {\displaystyle C^{\infty }(M)} -module of C ∞ ( M ) {\displaystyle C^{\infty }(M)} - multilinear forms This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to

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4704-412: Is given by all the smooth complex projective varieties . CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds . Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from

4816-414: Is not a tensor field, it still qualifies as a geometric object with a component-free interpretation. The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity . In electromagnetism , the electric and magnetic fields are combined into an electromagnetic tensor field . It

4928-436: Is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature , is written down. In the wake of the development of analytic geometry and plane curves, Alexis Clairaut began the study of space curves at just the age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to

5040-423: Is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport . An important example is provided by affine connections . For a surface in R , tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has

5152-417: Is the Levi-Civita connection of g {\displaystyle g} . In this case, ( J , g ) {\displaystyle (J,g)} is called a Kähler structure , and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold . A large class of Kähler manifolds (the class of Hodge manifolds )

5264-452: Is the Riemannian symmetric spaces , whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry . Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite . A special case of this is a Lorentzian manifold , which is

5376-467: Is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics . Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as

5488-477: Is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields . Beside the structure theory there is also the wide field of representation theory . Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. Gauge theory

5600-431: Is worth noting that differential forms , used in defining integration on manifolds, are a type of tensor field. In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus . Even to formulate such equations requires

5712-566: The Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity , and also to the Euler–Lagrange equations and the first theory of the calculus of variations , which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis . This theory

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5824-479: The Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and a year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing

5936-541: The Disquisitiones generales circa superficies curvas detailing the general theory of curved surfaces. In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and the inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map , Gaussian curvature , first and second fundamental forms , proved

6048-462: The Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory , and so their study is of considerable interest in physics. The apparatus of vector bundles , principal bundles , and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle . Loosely speaking, this structure by itself

6160-408: The Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field v k {\displaystyle v^{k}} transforms by the inverse Jacobian. A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber

6272-541: The Nijenhuis tensor (or sometimes the torsion ). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas . An almost Hermitian structure is given by an almost complex structure J , along with a Riemannian metric g , satisfying the compatibility condition An almost Hermitian structure defines naturally a differential two-form The following two conditions are equivalent: where ∇ {\displaystyle \nabla }

6384-501: The Poincaré conjecture . During this same period primarily due to the influence of Michael Atiyah , new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten , the only physicist to be awarded

6496-516: The Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds

6608-558: The Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces. At this time Gauss was already of the opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated

6720-587: The Weyl tensor providing insight into conformal geometry , and first defined the notion of a gauge leading to the development of gauge theory in physics and mathematics . In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of gauge theory and Yang–Mills theory in physics brought bundles and connections into focus, leading to developments in gauge theory . Many analytical results were investigated including

6832-455: The chain rule in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields. Abstractly, we can identify the chain rule as a 1- cocycle . It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying functorial properties of tensor constructions to

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6944-407: The circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for the purposes of mapping the shape of the Earth. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy , although in a much simplified form. Namely, as far back as Euclid 's Elements it

7056-621: The curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem , conjectured by Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then

7168-459: The method of exhaustion to compute the areas of smooth shapes such as the circle , and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders. There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance . Before the development of calculus by Newton and Leibniz , the most significant development in

7280-406: The n -dimensional real coordinate space R n {\displaystyle \mathbb {R} ^{n}} may be subjected to arbitrary affine transformations : (with n -dimensional indices, summation implied ). A covariant vector, or covector, is a system of functions v k {\displaystyle v_{k}} that transforms under this affine transformation by

7392-531: The natural sciences . Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity , and subsequently by physicists in the development of quantum field theory and the standard model of particle physics . Outside of physics, differential geometry finds applications in chemistry , economics , engineering , control theory , computer graphics and computer vision , and recently in machine learning . The history and development of differential geometry as

7504-473: The 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton . At this time, the recent work of René Descartes introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat , Newton, and Leibniz began the study of plane curves and the investigation of concepts such as points of inflection and circles of osculation , which aid in

7616-648: The Christoffel symbols, both coming from G in Gravitation . Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames , leading in the world of physics to Einstein–Cartan theory . Following this early development, many mathematicians contributed to the development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles , Shiing-Shen Chern who introduced characteristic classes to

7728-463: The Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system. Complex differential geometry is the study of complex manifolds . An almost complex manifold is a real manifold M {\displaystyle M} , endowed with a tensor of type (1, 1), i.e.

7840-404: The ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to

7952-432: The analysis of stress and strain in material object, and in numerous applications in the physical sciences . As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If

8064-403: The beginning and through the middle of the 19th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with

8176-410: The catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear. An application of the theorem is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction. This is of practical use in construction, as well as in

8288-441: The chain rule itself; this is why they also are intrinsic (read, 'natural') concepts. What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, post hoc approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities

8400-458: The corresponding field concept is called a section of the bundle: for m varying over M , a choice of vector where V m is the vector space "at" m . Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces ) the whole apparatus explained at component-free treatment of tensors carries over in

8512-796: The cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there). In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on M as multilinear maps on vectors and covectors, we can regard general ( k , l ) tensor fields on M as C ( M )-multilinear maps defined on k copies of X ( M ) {\displaystyle {\mathfrak {X}}(M)} and l copies of Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} into C ( M ). Now, given any arbitrary mapping T from

8624-443: The direction perpendicular to the fold, an attribute desirable for eating pizza, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly with corrugated fiberboard and corrugated galvanised iron , and in some forms of potato chips as well. Differential geometry Since the late 19th century, differential geometry has grown into

8736-444: The directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of curvature and double curvature , essentially the notion of principal curvatures later studied by Gauss and others. Around this same time, Leonhard Euler , originally

8848-417: The earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature. In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of

8960-464: The existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity

9072-404: The foundations of topology . At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program . As part of this broader movement, the notion of a topological space was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of

9184-405: The level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p , a hyperplane distribution is determined by a nowhere vanishing 1-form α {\displaystyle \alpha } , which is unique up to multiplication by a nowhere vanishing function: A local 1-form on M is a contact form if

9296-595: The manifold a tensor in the space where V is the tangent space at that point and V is the cotangent space . See also tangent bundle and cotangent bundle . Given two tensor bundles E → M and F → M , a linear map A : Γ( E ) → Γ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of E ∗ ⊗ F {\displaystyle \scriptstyle E^{*}\otimes F} if and only if it satisfies A ( fs ) = fA ( s ), for each section s in Γ( E ) and each smooth function f on M . Thus

9408-399: The map has at least two fixed points. Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2 n + 1) -dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with

9520-411: The mathematical basis of Einstein's general relativity theory of gravity . Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric , that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M {\displaystyle M}

9632-516: The measurement of curvature . Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition d 2 y = 0 {\displaystyle d^{2}y=0} indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers , Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at

9744-449: The modern formalism of the subject in terms of tensors and tensor fields . The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to classical antiquity . In particular, much was known about the geometry of the Earth , a spherical geometry , in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated

9856-442: The more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann , and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry . Dubbed the single most important work in the history of differential geometry, in 1827 Gauss produced

9968-413: The natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms . Beside Lie algebroids , also Courant algebroids start playing a more important role. A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which

10080-414: The origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it does make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion. As an advanced explanation of the tensor concept, one can interpret

10192-602: The proof of the Atiyah–Singer index theorem . The development of complex geometry was spurred on by parallel results in algebraic geometry , and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow , which culminated in Grigori Perelman 's proof of

10304-582: The range. A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the Einstein–Hilbert action in general relativity. The most common example of a scalar 1-density is the volume element , which in the presence of a metric tensor g is the square root of its determinant in coordinates, denoted det g {\displaystyle {\sqrt {\det g}}} . The metric tensor

10416-417: The restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. If the distribution H can be defined by a global one-form α {\displaystyle \alpha } then this form is contact if and only if the top-dimensional form is a volume form on M , i.e. does not vanish anywhere. A contact analogue of

10528-400: The rule The list of Cartesian coordinate basis vectors e k {\displaystyle \mathbf {e} _{k}} transforms as a covector, since under the affine transformation e k ↦ A k i e i {\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}} . A contravariant vector is

10640-402: The set of infinitely-differentiable tensor fields on M . Thus, are the sections of the ( m , n ) tensor bundle on M that are infinitely-differentiable. A tensor field is an element of this set. There is another more abstract (but often useful) way of characterizing tensor fields on a manifold M , which makes tensor fields into honest tensors (i.e. single multilinear mappings), though of

10752-537: The space. Differential geometry is closely related to, and is sometimes taken to include, differential topology , which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations , otherwise known as geometric analysis . Differential geometry finds applications throughout mathematics and

10864-437: The specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding. In modern mathematical terminology, the theorem may be stated as follows: The Gaussian curvature of a surface is invariant under local isometry . A sphere of radius R has constant Gaussian curvature which is equal to 1/ R . At the same time, a plane has zero Gaussian curvature. As

10976-401: The straight line paths on his map. Mercator noted that the praga were oblique curvatur in this projection. This fact reflects the lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of Gauss . The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around

11088-446: The study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed by Sophus Lie and Jean Gaston Darboux , leading to important results in the theory of Lie groups and symplectic geometry . The notion of differential calculus on curved spaces was studied by Elwin Christoffel , who introduced

11200-402: The subject and began the study of complex manifolds , Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms , Charles Ehresmann who introduced the theory of fibre bundles and Ehresmann connections , and others. Of particular importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced

11312-432: The subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with

11424-466: The systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann, was the development of an idea of Gauss's about the linear element d s {\displaystyle ds} of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into

11536-456: The tangent bundle TM = T ( M ) might sometimes be written as to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M . This should not be confused with the very similar looking notation in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M . Curly (script) letters are sometimes used to denote

11648-418: The tensor product of V with L , and consider tensor density fields with weight s . Half-densities are applied in areas such as defining integral operators on manifolds, and geometric quantization . When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M , we get back to a situation where a tensor field is synonymous with a tensor 'sitting at

11760-419: The theory of absolute differential calculus and tensor calculus . It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry . The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on

11872-456: The theory of plane curves, surfaces, and studied surfaces of revolution and envelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation. The field of differential geometry became an area of study considered in its own right, distinct from

11984-400: The time, later collated by L'Hopital into the first textbook on differential calculus , the tangents to plane curves of various types are computed using the condition d y = 0 {\displaystyle dy=0} , and similarly points of inflection are calculated. At this same time the orthogonality between the osculating circles of a plane curve and the tangent directions

12096-457: The understanding of differential geometry came from Gerardus Mercator 's development of the Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between praga , the lines of shortest distance on the Earth, and the directio ,

12208-482: The work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium , to the effect that Gaussian curvature is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there

12320-410: Was a non-Euclidean geometry, an elliptic geometry . The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift , On the hypotheses which lie at the foundation of geometry . In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began

12432-477: Was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the Earth leads to the conclusion that great circles , which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered

12544-551: Was used by Lagrange , a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem . Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to

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