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Concept in mathematics
geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or
Normal_bundle
branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There
Stable_normal_bundle
Scheme in 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
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Neighborhood of a submanifold
submanifold of a smooth manifold is an open set around it resembling the normal bundle. The tubular neighbourhood deformation retracts onto the submanifold
Tubular_neighborhood
Differentiable function whose derivative is everywhere injective
this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold. The stable normal bundle is
Immersion_(mathematics)
Concept in geometric topology
closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from
Normal_invariant
Technique in photogrammetry and computer vision
systems termed the normal equations. When solving the minimization problems arising in the framework of bundle adjustment, the normal equations have a sparse
Bundle_adjustment
Topics referred to by the same term
extension), used heavily in cryptography Normal bundle Normal cone, of a subscheme in algebraic geometry Normal coordinates, in differential in geometrical
Normal
Heart block in the right ventricle
A right bundle branch block (RBBB) is a heart block in the right bundle branch of the electrical conduction system. During a right bundle branch block
Right_bundle_branch_block
Line or vector perpendicular to a curve or a surface
normal vector Normal bundle – Concept in mathematics Pseudovector – Physical quantity that changes sign with improper rotation Tangential and normal components –
Normal_(geometry)
Topological space that locally resembles Euclidean space
and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere Sn is a generalisation of
Manifold
Invariant of a quadratic form over a field of characteristic 2
\partial M)} , we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding M ↪ D 4 {\displaystyle
Arf_invariant
Long exact sequence
i': X' = X ×Y Y' → Y' the induced map. Let N be the pullback of the normal bundle of i to X'. Then the refined Gysin homomorphism i! refers to the composition
Gysin_homomorphism
Complete manifolds of non-negative sectional curvature largely reduce to the compact case
closed totally convex, totally geodesic embedded submanifold whose normal bundle is diffeomorphic to M. Such a submanifold is called a soul of (M, g)
Soul_theorem
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Geometric plane containing the normal vector of a given surface
is zero, the surface is called a minimal surface. Earth normal section Normal bundle Normal curvature Osculating plane Principal curvature Tangent plane
Normal_plane_(geometry)
Anatomical cardiac structure
between the superior vena cava and the ascending aorta. Bachmann's bundle is, during normal sinus rhythm, the preferential path for electrical activation of
Bachmann's_bundle
Quadratic form related to curvatures of surfaces
case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by I I ( v , w ) = ( ∇ v w ) ⊥ , {\displaystyle
Second_fundamental_form
Fundamental formulas linking the metric and curvature tensor of a manifold
connection in the normal bundle. There are thus a pair of connections: ∇, defined on the tangent bundle of M; and D, defined on the normal bundle of M. These
Gauss–Codazzi_equations
Way to create new manifolds out of disk bundles
disk bundles. It was first described by John Milnor (1956) and subsequently used extensively in surgery theory to produce manifolds and normal maps with
Plumbing_(mathematics)
Collection of heart muscle cells
The bundle of His (BH) or His bundle (HB) (/hɪs/ "hiss") is a collection of heart muscle cells specialized for electrical conduction. As part of the electrical
Bundle_of_His
Russian mathematician (born 1966)
has a compact nonnegatively curved submanifold, called a soul, whose normal bundle is diffeomorphic to the original space. From the perspective of homotopy
Grigori_Perelman
Type of geometric transformation
normal bundle of Z {\displaystyle Z} in X {\displaystyle X} . Since E {\displaystyle E} is a smooth divisor (which has co-dim 1), its normal bundle is
Blowing_up
Way to join two given mathematical manifolds together
N_{M_{1}}V\setminus V\to N_{M_{2}}V\setminus V\cong N_{2}\setminus V,} where each normal bundle N M i V {\displaystyle N_{M_{i}}V} is diffeomorphically identified with
Connected_sum
Type of differentiable manifold
trivialisation of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle. A related
Parallelizable_manifold
Geometry formula
( F ) {\displaystyle e_{T}(F)} is the equivariant Euler form of the normal bundle of F {\displaystyle F} . The formula allows one to compute the equivariant
Localization formula for equivariant cohomology
Localization_formula_for_equivariant_cohomology
Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X
Vector_bundle
Digital storefront company selling video games and e-books
Humble Bundle, Inc. is a digital storefront for video games, which grew out of its original offering of Humble Bundles, collections of games sold at a
Humble_Bundle
Topological invariant in knot theory
a and b respectively in the positive and negative directions of the normal bundle to S. Given a basis b 1 , . . . , b 2 g {\displaystyle b_{1},...,b_{2g}}
Signature_of_a_knot
Invariant of framed knots
framing is a choice of a non-zero section in the normal bundle of the knot, i.e. a (non-zero) normal vector field. Given a framed knot C, the self-linking
Self-linking_number
Characteristic class in algebraic topology
of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the
Todd_class
Problem in algebraic geometry
before. Let F be the excess bundle of i and i'; that is, it is the pullback to X″ of the quotient of N by the normal bundle of i'. Let e(F) be the Euler
Residual_intersection
Restriction of electrical impulse flow in the heart's bundle branches
A bundle branch block is a partial or complete interruption in the flow of electrical impulses in either of the bundle branches of the heart's electrical
Bundle_branch_block
Concept in mathematics
scheme) of the first is the projective line bundle P(A + I) and has no section with trivial normal bundle (a section corresponds to a short exact sequence
Reductive_group
Algebraic structure used in topology
an orientation, a closed submanifold of X with an orientation on its normal bundle determines a cohomology class on X. If X is a noncompact manifold, then
Cohomology
Algebra in algebraic topology
Let ν Y / X → Y {\displaystyle \nu _{Y/X}\to Y} be the associated Normal bundle to this immersion. The Steenrod squares of α {\displaystyle \alpha }
Steenrod_algebra
Construct in algebraic geometry
complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes
Cotangent_complex
denote the Segre class of the normal cone to Z ↪ X {\displaystyle Z\hookrightarrow X} . For a holomorphic vector bundle E {\displaystyle E} over a complex
Segre_class
Topics referred to by the same term
containing the normal vector of a surface; see Normal plane (geometry). A term involving gears; see list of gear nomenclature. Normal bundle Normal section This
Normal_plane
Concept in differential geometry
orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures
Spin_structure
American mathematician (1940–2020)
TX: Publish or Perish. ISBN 978-0-914098-32-4. MR 2761185. Stable normal bundle Spivak pronoun Michael Spivak at the Mathematics Genealogy Project "1985
Michael_Spivak
Generalization of vector bundles
TX|_{Y}\to N_{Y/X}\to 0,} which can be used as a definition of the normal bundle N Y / X {\displaystyle N_{Y/X}} to Y {\displaystyle Y} in X {\displaystyle
Coherent_sheaf
sheaf of X in Y, then the normal sheaf, the dual of I / I 2 {\displaystyle I/I^{2}} , is locally free (thus a vector bundle) and the natural map Sym
Regular_embedding
Topological space associated to a vector bundle
and differential topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows
Thom_space
Branch of differential geometry
totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S (S is called the soul of M.) In particular, if M has strictly positive
Riemannian_geometry
Euler class of the cokernel bundle E / E ′ {\displaystyle E/E'} over X {\displaystyle X} . This bundle acts as the normal bundle of X {\displaystyle X} in
Virtual_fundamental_class
Characteristic class of oriented, real vector bundles
real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth
Euler_class
Generalizations of codimension-1 subvarieties of algebraic varieties
D is smooth, O D ( D ) {\displaystyle {\mathcal {O}}_{D}(D)} is the normal bundle of D in X. A Weil divisor D is said to be Cartier if and only if the
Divisor_(algebraic_geometry)
Medical condition
the left bundle branch means that it takes longer than normal for the left ventricle to fully depolarise. This can be due to a damaged bundle branch that
Left_bundle_branch_block
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
line bundles on X. 3. In general, O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} is the sheaf corresponding to a Weil divisor D (on a normal scheme)
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Characteristic classes of vector bundles
nonsingular quintic threefold in P 4 {\displaystyle \mathbb {P} ^{4}} . Its normal bundle is given by O X ( 5 ) {\displaystyle {\mathcal {O}}_{X}(5)} and we have
Chern_class
Relates the curvature of a Riemannian manifold to its topology
parallel vector field along the geodesic (i.e. a parallel section of the normal bundle to the geodesic), then Synge's earlier computation of the second variation
Synge's_theorem
Difference between the dimensions of mathematical object and a sub-object
the tangent bundle (the number of dimensions that you can move on the submanifold), the codimension is the dimension of the normal bundle (the number
Codimension
{\displaystyle j_{i}:V\hookrightarrow M_{i},} such that the Euler classes of the normal bundles are opposite: e ( N M 1 V ) = − e ( N M 2 V ) . {\displaystyle
Symplectic_sum
Well-behaved sequence in a commutative ring
complete intersection subscheme Y of any scheme X has a normal bundle which is a vector bundle, even though Y may be singular. Complete intersection ring
Regular_sequence
Topological space
submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric
Isoparametric_manifold
Description in Riemannian geometry
complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively
Sectional_curvature
Set of topological invariants
of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes
Stiefel–Whitney_class
Operator generalizing the Laplacian in differential geometry
_{S^{n-1}}f.} More generally, one can formulate a similar trick using the normal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically
Laplace–Beltrami_operator
Rigidity theorem in differential geometry
fundamental forms, there is also the (generally nontrivial) connection in the normal bundle which must be taken into account. In this generality, the fundamental
Bonnet_theorem
Generalization of a vector bundle
the normal cone to the closed scheme determined by I. If R = ⨁ 0 ∞ L ⊗ n {\displaystyle R=\bigoplus _{0}^{\infty }L^{\otimes n}} for some line bundle L
Cone_(algebraic_geometry)
Branch of algebraic geometry
formula says that A · B is represented by the top Chern class of the normal bundle of A in X. To give a definition, in the general case, of the intersection
Intersection_theory
Aspect of heart function
the right atrium to the atrioventricular node, along the bundle of His, and through the bundle branches to Purkinje fibers in the walls of the ventricles
Cardiac_conduction_system
S^{1}} -bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice. Normal bundle: associated
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
How spheres of various dimensions can wrap around each other
k-submanifolds of Sn+k which are "framed", i.e. have a trivialized normal bundle. Every map f : Sn+k → Sn is homotopic to a differentiable map with Mk
Homotopy_groups_of_spheres
Chinese-American mathematician (born 1949)
interpreted either as the mean curvature being parallel as a section of the normal bundle, or as the constancy of the length of the mean curvature. Under the
Shing-Tung_Yau
Maths
1])\cap \partial M=h(\partial F\times [-1,1])} . In other words, if its normal bundle is trivial. This means, for example that a curve in a surface is 2-sided
2-sided
Theorem relating Milnor K-theory and Galois cohomology
a morphism from the motivic sphere to the Thom space of the motivic normal bundle over a smooth projective algebraic variety. A construction of the motivic
Norm residue isomorphism theorem
Norm_residue_isomorphism_theorem
Key result in general relativity
from a connected n-dimensional manifold M into M which has a trivial normal bundle, one may consider the induced Riemannian metric g = f *g as well as
Positive_energy_theorem
Intrinsic geometric structures in mathematics
Unlike the more intuitive normal bundle, easily visualised as a tubular neighbourhood of an embedded surface in E3, the frame bundle is an intrinsic invariant
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Concept in algebraic geometry
} denotes the dual of a line bundle. Suppose that D is a smooth divisor on X. Its normal bundle extends to a line bundle O ( D ) {\displaystyle {\mathcal
Adjunction_formula
Concept in differential topology
surfaces embedded in S n + 2 {\displaystyle S^{n+2}} with trivialized normal bundle. Kervaire (1960) used his invariant for n = 10 to construct the Kervaire
Kervaire_invariant
Group of medical conditions characterized by irregular heartbeat
including when it is too fast or too slow. Essentially, this is anything but normal sinus rhythm. A resting heart rate that is too fast – above 100 beats per
Arrhythmia
Constructions in nonsmooth analysis
or more generally of the normal bundle of an embedded submanifold — to possibly non-smooth subsets of vector spaces. Normal cones provide, among other
Normal cone (variational analysis)
Normal_cone_(variational_analysis)
Microstructure of natural materials
of protection. Under stress, the Bouligand planes fail via normal bundle fracture or bundle separation mechanisms. The exocuticle-endocuticle interface
Bouligand_structure
Concept from algebraic geometry
relative Kähler differentials and N U / Z {\displaystyle N_{U/Z}} is the normal bundle to i {\displaystyle i} . For a smooth curve C, its dualizing sheaf ω
Dualizing_sheaf
Electrocardiogram waveform representing ventricular contraction in the heart
Depolarization of the heart ventricles occurs almost simultaneously, via the bundle of His and Purkinje fibers. If they are working efficiently, the QRS complex
QRS_complex
Study of systems of inequalitites
smooth submanifold of R n {\displaystyle \mathbb {R} ^{n}} with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset
Real_algebraic_geometry
Mathematical measure in Riemannian geometry
submanifold. Conversely the normal curvature is the norm of the projection of D T / d s {\displaystyle DT/ds} on the normal bundle to the submanifold at the
Geodesic_curvature
Medical condition
atrioventricular node (AV) node. In normal conduction, the impulse would travel across the bundle of His (AV bundle), down the bundle branches, and into the Purkinje
Sinoatrial_block
Concept in mathematics
is simpler for a vector bundle (i.e., a variety corresponding to a locally free sheaf of constant rank). We say a vector bundle E on an algebraic variety
Equivariant_sheaf
Mod 2 invariant of (4k+1)-dimensional manifold
) {\displaystyle v_{2k}\in H^{2k}(M;Z_{2})} is the Wu class of the normal bundle of M {\displaystyle M} and S q 1 {\displaystyle Sq^{1}} is the Steenrod
De_Rham_invariant
Abnormal heart rhythm due to faulty electrical connections in the heart
delay at the AV node, the stimulus travels through the bundle of His to the left and right bundle branches and then to the Purkinje fibers and the endocardium
Wolff–Parkinson–White syndrome
Wolff–Parkinson–White_syndrome
Heart condition
pacemaker-generated paced rhythm. Normal variation causing LAD is an age-related physiologic change. Conduction defects such as left bundle branch block or left anterior
Left_axis_deviation
Topological spaces whose union is a boundary
using the notion of X-structure (or G-structure). Very briefly, the normal bundle ν of an immersion of M into a sufficiently high-dimensional Euclidean
Cobordism
Calculus of functions generalization
. Indeed, the choice of metric makes the normal bundle ν i {\displaystyle \nu _{i}} a complementary bundle to T N {\displaystyle TN} ; i.e., T M | N
Calculus_on_Euclidean_space
Techniques in topology used to produce one finite-dimensional manifold from another
that a normal map of degree one to X exists if and only if the Spivak normal fibration of X has a reduction to a stable vector bundle. If normal maps of
Surgery_theory
Examination of the heart's electrical activity
through the bundle of His and bundle branches. After the Bundle of His, the conduction system splits into the left bundle branch and the right bundle branch
Electrocardiography
underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian subgroup with underlying
Tangent_Lie_group
Manifold union
framing of the attaching sphere, since it gives trivialization of its normal bundle. { 0 } j × S m − j − 1 ⊂ D j × D m − j = H j {\displaystyle \{0\}^{j}\times
Handle_decomposition
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
∈ H {\displaystyle [Y]\in H} is given by the global sections of the normal bundle N Y / X {\displaystyle N_{Y/X}} ; that is, T [ Y ] H = H 0 ( Y , N Y
Hilbert_scheme
Medical condition
nodes, the electrical signal travels through Bundle of His and divides into the right bundle and left bundle, which are located within the interventricular
Atrioventricular_block
Version without requiring the smooth manifolds involved to carry a complex structure
theorem via explicit computation for line bundles. If f: X → Y is an embedding, then the Thom space of the normal bundle of X in Y can be viewed as a tubular
Riemann–Roch theorem for smooth manifolds
Riemann–Roch_theorem_for_smooth_manifolds
Concept in algebraic geometry
an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to
Ample_line_bundle
tautological line bundle on projective space, and its d-th powers for d = 1, 2, 3, ... ; when V is non-singular, it is projectively normal if and only if
Homogeneous_coordinate_ring
Concept in algebraic geometry
geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described
Nef_line_bundle
Branch of topology
from L M × L M {\displaystyle LM\times LM} to the Thom space of the normal bundle of M a p ( S 1 ∨ S 1 , M ) {\displaystyle {\rm {Map}}(S^{1}\lor S^{1}
String_topology
Possibility of a consistent definition of "clockwise" in a mathematical space
also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle, so it is a fiber bundle with structure group GL ( n , R ) {\displaystyle
Orientability
Theorem in algebraic geometry
scheme Y, there is a more elementary description: the normal bundle of X in Y is a vector bundle of rank r, and the dualizing sheaf of X is given by: ω
Serre_duality
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