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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
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)
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
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)
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
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)
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)
Concept in mathematics
invariant differential operators, and spherical functions, American Mathematical Society ISBN 0821826735 Sigurdur Helgason (2011) Integral Geometry and Radon
Integral_geometry
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
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 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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
algebra), affine derived algebraic geometry over characteristic zero is equivalent to the theory of commutative differential graded rings. One of the main
Derived_scheme
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)
Concept in differential topology
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
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)
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
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
When differentials on an algebraic surface represent as a pullback of an algebraic curve
two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pullback of an
Castelnuovo–de Franchis theorem
Castelnuovo–de_Franchis_theorem
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
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
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
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)
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator
Elliptic_complex
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
cohomology and the intersection forms of 4-manifolds". Journal of Differential Geometry. 24 (3): 275–341. doi:10.4310/jdg/1214440551. ISSN 0022-040X. MR 0679066
Donaldson_invariant
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
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
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
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
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
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
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
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
Submanifold metric tensor
the pullback. It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:
Induced_metric
the Weyl tensor, one of the two primitive invariants in conformal differential geometry. Aside from the obstruction tensor, the ambient construction can
Ambient_construction
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)
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
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
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
UTM becomes a Sasakian manifold. Jeffrey M. Lee: Manifolds and Differential Geometry. Graduate Studies in Mathematics Vol. 107, American Mathematical
Unit_tangent_bundle
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
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
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
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
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
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
Long exact sequence
} induces a map in cohomology H ∗ {\displaystyle H^{\ast }} called its pullback π ∗ {\displaystyle \pi ^{\ast }} π ∗ : H ∗ ( M ) ⟶ H ∗ ( E ) . {\displaystyle
Gysin_homomorphism
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
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)
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
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
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
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
Maurer–Cartan form – Mathematical concept R. W. Sharpe (1996). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag
Darboux_derivative
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
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
Analogs of homology groups for algebraic varieties
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) of an algebraic variety over any field are algebro-geometric
Chow_group
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
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
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
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
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