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Mathematical operation
ones. Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors
Pullback (differential geometry)
Pullback_(differential_geometry)
Process in mathematics
Pullbacks can be applied to many other objects such as differential forms and their cohomology classes; see Pullback (differential geometry) Pullback
Pullback
Mathematics of smooth surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most
Differential geometry of surfaces
Differential_geometry_of_surfaces
field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
Expression that may be integrated over a region
under pullback. Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is
Differential_form
Mathematical notion of infinitesimal difference
mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. The term differential is used nonrigorously in calculus
Differential_(mathematics)
Linear approximation of smooth maps on tangent spaces
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle
Pushforward_(differential)
Most general completion of a commutative square given two morphisms with same codomain
mediating morphism u : Q → P above is not required to be unique. Pullbacks in differential geometry Join (relational algebra) Mitchell, p. 9 Lee, John M. (2003)
Pullback_(category_theory)
In geometry, a valuation is a finitely additive function from a collection of subsets of a set X {\displaystyle X} to an abelian semigroup. For example
Valuation_(geometry)
Generalizations of codimension-1 subvarieties of algebraic varieties
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common
Divisor_(algebraic_geometry)
In the mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties
Affine_geometry_of_curves
Fiber bundle induced by a map of its base space
construction is useful in differential geometry and topology. Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds
Pullback_bundle
Concept in mathematics
invariant differential operators, and spherical functions, American Mathematical Society ISBN 0821826735 Sigurdur Helgason (2011) Integral Geometry and Radon
Integral_geometry
differential system ( M , I ) {\displaystyle (M,I)} consists of a submanifold N ⊂ M {\displaystyle N\subset M} having the property that the pullback to
Differential_ideal
Type of derivative in differential geometry
In differential geometry, the Lie derivative (/liː/ LEE), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including
Lie_derivative
Concept in differential geometry
a set equipped with a diffeology. Many of the standard tools of differential geometry extend to diffeological spaces, which beyond manifolds include arbitrary
Diffeology
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: Glossary of
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Aspect of theoretical physics
defines a Riemannian metric called the quantum metric (equivalently, the pullback of the Fubini–Study metric on projective Hilbert space), while the imaginary
Quantum geometry (condensed matter)
Quantum_geometry_(condensed_matter)
Topics referred to by the same term
topology Pullback (differential geometry), a term in differential geometry Pullback (category theory), a term in category theory Pullback attractor, an aspect
Pull_back_(disambiguation)
glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Foundational result in symplectic geometry
In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms,
Darboux's_theorem
Type of manifold in differential geometry
In differential geometry, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate differential 2-form ω
Symplectic_manifold
Infinitesimal version of Lie groupoid
13–15. Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge: Cambridge
Lie_algebroid
Generalization of an ordered basis of a vector space
in conjunction with an origin) often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. In lay terms
Moving_frame
Inclusion of one mathematical structure in another, preserving properties of interest
[1993]. Differential manifolds. Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8. Lang, Serge (1999). Fundamentals of Differential Geometry. Graduate
Embedding
vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with
Vector-valued differential form
Vector-valued_differential_form
Mathematical description of spacetime used in relativity
which is formulated in the mathematics of differential geometry of differential manifolds. When this geometry is used as a model of spacetime, it is known
Minkowski_spacetime
Construct allowing differentiation of tangent vector fields of manifolds
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent
Affine_connection
Natural moving frame in differential geometry of surfaces
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame
Darboux_frame
Concept in mathematics
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming
Normal_bundle
transformations of the manifold (the action of the transformation on differential forms is just the pullback). More generally, the exterior derivative d : Ω n
Invariant differential operator
Invariant_differential_operator
Scheme in algebraic geometry
algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. The
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Branch of mathematics
to other structures of differential geometry; the assimilation of the Kodaira–Spencer theory into the abstract algebraic geometry of Grothendieck, with
Deformation_(mathematics)
Mathematical result in differential geometry
differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator
Atiyah–Singer_index_theorem
Every Riemannian manifold can be isometrically embedded into some Euclidean space
2022-05-06. Kobayashi, Shoshichi; Nomizu, Katsumi (1969). Foundations of differential geometry. Vol II. Interscience Tracts in Pure and Applied Mathematics. Vol
Nash_embedding_theorems
Exterior algebraic map taking tensors from p forms to n-p forms
play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows
Hodge_star_operator
Smooth manifold with an inner product on each tangent space
In differential geometry, a Riemannian manifold (or Riemann space) is a geometric space on which many geometric notions such as distance, angles, length
Riemannian_manifold
Dual space to the tangent space in differential geometry
In differential geometry, the cotangent space is a vector space associated with a point x {\displaystyle x} on a smooth (or differentiable) manifold M
Cotangent_space
Construction for vector bundles
In differential geometry, the determinant line bundle is a construction, which assigns every vector bundle over paracompact spaces a line bundle. Its
Determinant_line_bundle
Structure defining distance on a manifold
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface)
Metric_tensor
algebra), affine derived algebraic geometry over characteristic zero is equivalent to the theory of commutative differential graded rings. One of the main
Derived_scheme
Generalization of affine connections
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also
Cartan_connection
every point p ∈ N {\displaystyle \textstyle p\in N} is annihilated by (the pullback of) each α i {\displaystyle \textstyle \alpha _{i}} . A maximal integral
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric
Bundle_metric
UTM becomes a Sasakian manifold. Jeffrey M. Lee: Manifolds and Differential Geometry. Graduate Studies in Mathematics Vol. 107, American Mathematical
Unit_tangent_bundle
algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings
Algebraic_cycle
Tensorial object depending on two points in a manifold
In differential geometry and general relativity, a bitensor (or bi-tensor) is a tensorial object that depends on two points in a manifold, as opposed
Bitensor
Characteristic classes of vector bundles
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated
Chern_class
Algebra associated to any vector space
algebra of differential forms on a manifold the structure of a differential graded algebra. The exterior derivative commutes with pullback along smooth
Exterior_algebra
Diophantine geometry Glossary of classical algebraic geometry Glossary of differential geometry and topology Glossary of Riemannian and metric geometry List
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Distance-preserving mathematical transformation
(1969). Introduction to Geometry, Second edition. Wiley. ISBN 9780471504580. Lee, Jeffrey M. (2009). Manifolds and Differential Geometry. Providence, RI: American
Isometry
Concept in mathematics
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle;
Connection_(principal_bundle)
Math/physics concept
specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms
Connection_form
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Manifold upon which it is possible to perform calculus
The study of calculus on differentiable manifolds is known as differential geometry. "Differentiability" of a manifold has been given several meanings
Differentiable_manifold
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space
Hilbert_scheme
Module over a sheaf of differential operators
algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators
D-module
Affine connection on the tangent bundle of a manifold
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine
Levi-Civita_connection
Mathematical parametrization of vector spaces by another space
morphism over X 1 {\displaystyle X_{1}} from E 1 {\displaystyle E_{1}} to the pullback bundle g ∗ E 2 {\displaystyle g^{*}E_{2}} . Given a vector bundle π: E
Vector_bundle
Mathematical concept
Supérieure. 21: 153–206. doi:10.24033/asens.538. R. W. Sharpe (1996). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag
Maurer–Cartan_form
System of moving vectors in differential geometry
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If
Parallel_transport
Structure group sub-bundle on a tangent frame bundle
In differential geometry, a G-structure on an n {\displaystyle n} -manifold M {\displaystyle M} , for a given structure group G {\displaystyle G} , is
G-structure_on_a_manifold
Cohomology with real coefficients computed using differential forms
integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples
De_Rham_cohomology
Internal groupoid in the category of smooth manifolds
ISBN 978-1-4008-8173-4. Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge: Cambridge
Lie_groupoid
German-American mathematician (1928–1999)
MR 0288405. Zbl 0227.35016. Moser, J. (1973). "On a nonlinear problem in differential geometry". In Peixoto, M. M. (ed.). Dynamical systems. Symposium held at
Jürgen_Moser
Mathematical condition
mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is
Poincaré_lemma
Array of numbers describing a metric connection
a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric
Christoffel_symbols
Mathematical construct of fiber bundles
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of
Solder_form
Tool to track locally defined data attached to the open sets of a topological space
several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold
Sheaf_(mathematics)
Approach to general relativity
Each covector is a solder form. From the point of view of the differential geometry of fiber bundles, the n vector fields { e a } a = 1 … n {\displaystyle
Tetrad_formalism
Association of cohomology classes to principal bundles
unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in
Characteristic_class
Vector bundle of cotangent spaces at every point in a manifold
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every
Cotangent_bundle
Differential geometry technique
In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up
Cartan's_equivalence_method
Concept in algebraic geometry
of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle
Ample_line_bundle
Differentiable function whose derivative is everywhere injective
Applicable differential geometry, Cambridge, England: Cambridge University Press, ISBN 978-0-521-23190-9 Darling, Richard William Ramsay (1994), Differential forms
Immersion_(mathematics)
Mathematical concept
Riemannian geometry, Walter de Greuter, ISBN 978-3-11-008673-7. Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Volume
Complex_projective_space
Transformation that preserves area measure of regions
In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of
Equiareal_map
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator
Elliptic_complex
Map from multiple vectors to an underlying field of scalars, linear in each argument
To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking
Multilinear_form
Algebraic structure used in topology
many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces X {\displaystyle X} and Y {\displaystyle
Cohomology
Generalization of vector bundles
_{\mathbb {P} ^{n}}\to \Omega _{X}\to 0} where the second map is the pullback of differential forms, and the first map sends ϕ ↦ d ( f ⋅ ϕ ) {\displaystyle \phi
Coherent_sheaf
Differential map between manifolds whose differential is everywhere surjective
differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion
Submersion_(mathematics)
Projective variety that is also an algebraic group
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety
Abelian_variety
Fiber bundle whose fibers are group torsors
Principal bundles have important applications in topology and differential geometry and mathematical gauge theory. They have also found application
Principal_bundle
Exceptional functor
Daniel Henry (1975), "Fibre bundles and the Euler characteristic" (PDF), Journal of Differential Geometry, 10 (1): 39–48, doi:10.4310/jdg/1214432674
Shriek_map
Vector bundle of rank 1
X {\displaystyle X} to P r {\displaystyle \mathbf {P} ^{r}} , and the pullback of the dual of the tautological bundle under this map is L {\displaystyle
Line_bundle
Gradient flow of the Yang–Mills action functional
In differential geometry, the Yang–Mills flow is a gradient flow described by the Yang–Mills equations, hence a method to describe a gradient descent
Yang–Mills_flow
Differential geometry construct on fiber bundles
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of
Ehresmann_connection
η−1F) on Y; in many cases where the Penrose transform is of interest, this pullback turns out to be an isomorphism. One then pushes the resulting cohomology
Penrose_transform
Canonical differential form
form. The tautological one-form is the unique one-form that "cancels" pullback. That is, let β {\displaystyle \beta } be a 1-form on Q . {\displaystyle
Tautological_one-form
Intrinsic geometric structures in mathematics
in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Concept in differential geometry
In differential geometry, a complete Riemannian manifold ( M , g ) {\displaystyle (M,g)} is called a Ricci soliton if, and only if, there exists a smooth
Ricci_soliton
Unsolved problem in geometry
the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex
Hodge_conjecture
Book by Michael Spivak
concepts and constructions (e.g., tensor products, differential forms, tangent spaces, pullbacks, exterior derivatives, cube and chains) are introduced
Calculus_on_Manifolds_(book)
Mathematical technique for vector bundles
{\displaystyle p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)} is injective, and the pullback bundle p ∗ ξ : p ∗ E → Y {\displaystyle p^{*}\xi \colon p^{*}E\rightarrow
Splitting_principle
Degree of differentiability of a function or map
a smooth manifold M {\displaystyle M} , plays a central role in differential geometry: many geometric objects on M {\displaystyle M} can be described
Smoothness
Concept in differential geometry
In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills–Higgs (YMH) pair is a Yang–Mills–Higgs pair around which the Yang–Mills–Higgs
Stable_Yang–Mills–Higgs_pair
singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic
Schwarz's_list
Homeomorphism between plane domains
conformal. Without appeal to an auxiliary metric, consider the effect of the pullback under f of the usual Euclidean metric. The resulting metric is then given
Quasiconformal_mapping
Differential operator in mathematics
and Riemannian and pseudo-Riemannian manifold. The Laplacian in differential geometry. The discrete Laplace operator is a finite-difference analog of
Laplace_operator
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
Boy/Male
Greek
Greek surname. Euclid was an early developer of geometry theories.
Boy/Male
Irish
From the Latin patricius “â€nobly born.â€â€ The patron saint of Ireland, it is hard to differentiate between fact and myth. What is probably true is that he was born in Britain around 373 AD and was brought to Ireland as a slave at the age of seven, possibly by Niall of the Nine Hostages (read the legend). Forced to guard sheep on the Slemish Mountains in Country Antrim for six years he had a vision urging him to convert his captors. He escaped to France where he trained as a priest before returning to Ireland where he banished the snakes (i.e. paganism) and converted the population to Christianity. Both Patrick and Padraig are very popular names in Ireland.
Boy/Male
Irish
From the Latin patricius “â€nobly born.â€â€ The patron saint of Ireland, it is hard to differentiate between fact and myth. What is probably true is that he was born in Britain around 373 AD and was brought to Ireland as a slave at the age of seven, possibly by Niall of the Nine Hostages (read the legend). Forced to guard sheep on the Slemish Mountains in Country Antrim for six years he had a vision urging him to convert his captors. He escaped to France where he trained as a priest before returning to Ireland where he banished the snakes (i.e. paganism) and converted the population to Christianity. Both Patrick and Padraig are very popular names in Ireland.
Boy/Male
British, English
Crown
Boy/Male
Afghan, Arabic, Muslim, Pashtun
One who can Differentiate; Comely; One who Distinguishes Truth from Falsehood
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
Boy/Male
American, Australian, British, Christian, English, French, German
Rufus; Red-haired; Red Skinned; Little Red One
Male
Russian
(Феофилакт) Russian form of Greek Theophylaktos, FEOFILAKT means "God-guard."
Boy/Male
Indian, Sanskrit
Installed as King
Girl/Female
Bengali, Hindu, Indian, Sanskrit, Tamil, Telugu, Traditional
Prize; Garland of Lord Vishnu
Boy/Male
Indian
Arbitrator, Judge
Female
Basque
, acorn.
Female
English
Anglicized form of Hebrew Shelomiyth, SHELOMITH means "peaceful." In the bible, this is the name of many characters, including a daughter of Dibri of the tribe of Dan.
Boy/Male
Tamil
Sambhavan | ஸமபாவந
Respect, Honour, Possibility
Girl/Female
Muslim
Purity, Righteousness, Honesty
Boy/Male
American, Chinese, Christian, German
Strong; Manly
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
PULLBACK DIFFERENTIAL-GEOMETRY
n.
An increment, usually an indefinitely small one, which is given to a variable quantity.
n.
The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point.
a.
Ready to obey; reverent; differential; also, servilely submissive.
a.
Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.
pl.
of Differentia
n.
The pollack.
n.
That which holds back, or causes to recede; a drawback; a hindrance.
n.
The quillback.
v. t.
To distinguish or mark by a specific difference; to effect a difference in, as regards classification; to develop differential characteristics in; to specialize; to desynonymize.
n.
A marine gadoid food fish of Europe (Pollachius virens). Called also greenfish, greenling, lait, leet, lob, lythe, and whiting pollack.
n.
A small difference in rates which competing railroad lines, in establishing a common tariff, allow one of their number to make, in order to get a fair share of the business. The lower rate is called a differential rate. Differentials are also sometimes granted to cities.
n.
A characteristic or essential attribute; a differential.
a.
That deduces; inferential.
a.
Of or pertaining to a differential, or to differentials.
n.
One of two coils of conducting wire so related to one another or to a magnet or armature common to both, that one coil produces polar action contrary to that of the other.
n.
An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.
v. t.
To obtain the differential, or differential coefficient, of; as, to differentiate an algebraic expression, or an equation.
n.
A form of conductor used for dividing and distributing the current to a series of electric lamps so as to maintain equal action in all.
a.
Relating to differences of motion or leverage; producing effects by such differences; said of mechanism.
n.
The quillback.