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Normal bundle

In differential geometry, the normal bundle of a submanifold Y embedded in a smooth manifold X via an inclusion map i: Y \hookrightarrow X is defined as the quotient vector bundle N_{Y/X} = i^*(TX) / TY over Y, where TX is the tangent bundle of X and the fiber at each point y \in Y consists of equivalence classes of tangent vectors to X at i(y) modulo those tangent to Y.^[1] This structure encodes the transverse directions to Y within X, providing a canonical way to describe infinitesimal deformations perpendicular to the submanifold. When X is equipped with a Riemannian metric, the normal bundle admits an orthogonal identification with the subbundle of i^*(TX) consisting of vectors perpendicular to TY at each point, forming the orthogonal complement TY^\perp \subset i^*(TX) such that i^*(TX)_y \cong T_y Y \oplus T_y Y^\perp.^[2] This identification relies on the metric's inner product and ensures the normal bundle is a smooth vector bundle of rank equal to \dim X - \dim Y. Key properties include its compatibility with parallel transport, preserving orthogonality along geodesics, and its role in decomposing the ambient tangent space as a direct sum of tangent and normal components.^[2] The normal bundle is fundamental in several areas of geometry and topology. By the tubular neighborhood theorem, for a compact submanifold Y \subset X, there exists an open neighborhood of Y in X diffeomorphic to the total space of a disk bundle in N_{Y/X}, allowing local coordinates where Y is modeled as the zero section.^[3] This theorem facilitates the study of embeddings, intersections, and deformations. Additionally, through the Gauss–Weingarten equations, the normal bundle connects to the second fundamental form, which measures the extrinsic curvature of Y in X via the shape operator mapping tangent vectors to normal directions.^[2] In topology, normal bundles classify stable embeddings and appear in surgery theory, while in algebraic geometry, analogous constructions arise for subschemes in varieties.^[1]^[4]

Definition

Riemannian manifolds

In a Riemannian manifold (M, g) with a submanifold S \subset M, the normal space N_p S at a point p \in S is defined as the orthogonal complement of the tangent space T_p S in the tangent space T_p M with respect to the Riemannian metric g.^[5] Specifically, N_p S = \{ v \in T_p M \mid g(v, w) = 0 \ \forall w \in T_p S \}, which ensures a direct sum decomposition T_p M = T_p S \oplus N_p S.^[5] The normal bundle NS is constructed as the disjoint union NS = \bigcup_{p \in S} N_p S, equipped with the natural projection \pi: NS \to S given by \pi(v) = p for v \in N_p S, forming a smooth vector bundle of rank \dim M - \dim S over S.^[5] Local trivializations of NS are obtained using adapted orthonormal frames on neighborhoods of points in S, where the frame spans T_q S with the remaining vectors spanning the normal space, ensuring smooth transition functions across overlaps.^[5] The Riemannian metric g on M induces an inner product on each fiber N_p S by restriction, defined as \langle v, w \rangle_{N_p S} = g(v, w) for v, w \in N_p S, making NS a Riemannian vector bundle.^[5] This fiberwise inner product is smooth in p and compatible with the bundle structure, allowing for orthonormal frames in the normal directions. For example, the metric enables parallel transport of vectors in the normal bundle along geodesics perpendicular to S; specifically, the normal exponential map \exp^\perp: NS \to M, which sends v \in N_p S to the endpoint of the geodesic starting at p with initial velocity v (initially normal to T_p S), preserves lengths and angles via the Levi-Civita connection, transporting normal vectors isometrically along these radial geodesics.^[5]

General immersions

Let i: N \to M be a smooth immersion between smooth manifolds of dimensions \dim N = n and \dim M = m, with m \geq n. The pullback bundle i^* TM is the vector bundle over N whose fiber over p \in N is T_{i(p)} M. The differential di: TN \to i^* TM is a smooth bundle morphism that is injective on each fiber, so its image \operatorname{im}(di) is a smooth subbundle of i^* TM isomorphic to TN. The normal bundle of the immersion, denoted T_{M/N} or \nu(i), is the quotient bundle (i^* TM) / \operatorname{im}(di) over N, where the quotient is taken fiberwise.^[6]^[7] This yields the short exact sequence of vector bundles over N:

$0 \to TN \xrightarrow{di} i^* TM \to T_{M/N} \to 0,

where the first map is the inclusion via di, and the second is the canonical projection onto the quotient. The sequence is exact at TN because di is fiberwise injective, exact at i^* TM because \operatorname{im}(di) is the kernel of the projection, and exact at T_{M/N} by the definition of the quotient. Each fiber of T_{M/N} over p \in N is thus T_{i(p)} M / di_p(T_p N), a vector space of dimension m - n, so T_{M/N} is a smooth vector bundle of rank m - n. Since short exact sequences of vector bundles over paracompact manifolds (such as smooth manifolds) always split, there exists a bundle isomorphism i^* TM \cong TN \oplus T_{M/N} over N, though the splitting is not canonical without further structure.^[7] The construction applies uniformly to both immersions and embeddings. For an embedding, the image i(N) is an embedded submanifold diffeomorphic to N, and the normal bundle may equivalently be viewed as the quotient TM|_{i(N)} / T i(N) over the image submanifold.^[6] Given a linear connection \nabla on TM, the pullback induces a connection on i^* TM. If this connection preserves the subbundle \operatorname{im}(di), it further induces a normal connection \nabla^\perp on the quotient bundle T_{M/N} by setting \nabla^\perp_X \xi = \pi(\nabla_X \tilde{\xi}) for X \in \Gamma(TN), \xi \in \Gamma(T_{M/N}), a lift \tilde{\xi} \in \Gamma(i^* TM) of \xi, and projection \pi: i^* TM \to T_{M/N}; this is well-defined and satisfies the axioms of a linear connection independently of the choice of lift.^[8]

Conormal bundle

In differential geometry, for an immersed submanifold Y of a smooth manifold X, the conormal bundle T^*_{X/Y} is defined as the annihilator of the tangent bundle TY in the restriction of the cotangent bundle T^*X|_Y, consisting of all covectors in T^*X|_Y that vanish on vectors tangent to Y.^[9] This makes T^*_{X/Y} a subbundle of T^*X|_Y with rank equal to the codimension of Y in X. The conormal bundle fits into a short exact sequence of vector bundles over Y:

$0 \to T^*_{X/Y} \to T^*X|_Y \to T^*Y \to 0,

which is the dual of the exact sequence defining the normal bundle T_{X/Y}.^[10] This sequence arises from the restriction of the cotangent bundle and the annihilation property, ensuring that the quotient T^*X|_Y / T^*_{X/Y} \cong T^*Y. In the algebraic geometry setting, where Y is a subvariety of a smooth variety X, the conormal sheaf \mathcal{I}_Y / \mathcal{I}_Y^2—with \mathcal{I}_Y \subset \mathcal{O}_X the ideal sheaf of Y—underlies the conormal bundle when Y is smooth.^[11] This sheaf fits into the exact sequence of sheaves on Y:

$0 \to \mathcal{I}_Y / \mathcal{I}_Y^2 \to \Omega^1_X|_Y \to \Omega^1_Y \to 0,

capturing infinitesimal deformations transverse to Y.^[12] The conormal bundle is naturally isomorphic to the dual of the normal bundle: T^*_{X/Y} \cong (T_{X/Y})^*.^[10] This duality highlights its role as the cotangent counterpart to the normal bundle, facilitating computations in deformation theory and intersection homology.

Properties and Constructions

Stable normal bundle

The stable normal bundle of a smooth manifold M, often denoted \tilde{\nu}_M, is defined by stabilizing the normal bundle \nu_M associated to an embedding i: M \hookrightarrow \mathbb{R}^k for sufficiently large k. Specifically, \tilde{\nu}_M is the stable equivalence class of \nu_M \oplus \varepsilon^{k - \dim M}, where \varepsilon^l denotes the trivial real vector bundle of rank l over M. This stabilization ensures that the resulting bundle is independent of the choice of embedding up to stable isomorphism, as different embeddings yield stably equivalent normal bundles.^[13] The Whitney embedding theorem guarantees that any smooth n-dimensional manifold M admits an embedding into \mathbb{R}^{2n}, providing a concrete realization of the stable normal bundle as the orthogonal complement to the tangent bundle in the trivial bundle \varepsilon^{2n}. This theorem, proved by Hassler Whitney in 1944, establishes that the stable normal bundle is well-defined as a stable class in the orthogonal group O, represented by a map \tilde{\nu}_M: M \to [BO](/page/Bo) up to homotopy, where BO is the classifying space for real vector bundles. Consequently, the stable normal bundle captures the topological embedding properties of M in high-dimensional Euclidean space without dependence on the precise dimension beyond stabilization.^[13] In vector bundle theory, two bundles \xi and \eta over the same base are stably equivalent if there exist integers m, l \geq 0 such that \xi \oplus \varepsilon^m \cong \eta \oplus \varepsilon^l. This equivalence relation groups bundles into stable classes, forming the stable orthogonal group, and the stable normal bundle \tilde{\nu}_M belongs to this structure, with its classifying map to BO invariant under stabilization. Stable equivalence preserves characteristic classes, such as Stiefel-Whitney classes, ensuring that \tilde{\nu}_M provides a canonical invariant for M.^[13] In oriented cobordism theory, the stable normal bundle plays a pivotal role in the Pontryagin-Thom construction, which identifies the oriented cobordism group \Omega_n^{SO} with the nth homotopy group of the Thom spectrum MSO. For an oriented n-manifold M, the classifying map of the oriented stable normal bundle \tilde{\nu}_M: M \to BSO composes with the Thom map to yield the Thom space Th(\tilde{\nu}_M), and the Pontryagin-Thom collapse map from the one-point compactification of the embedding realizes the fundamental class of M in homotopy terms, determining its cobordism class via the oriented structure on \tilde{\nu}_M. This construction, due to René Thom and Lev Pontryagin, reduces cobordism computations to stable homotopy theory.^[14]

Relation to the tangent bundle

The relation between the normal bundle \nu of a submanifold N in a manifold M and the tangent bundles is captured by the short exact sequence of vector bundles

$0 \to TN \to TM|_N \to \nu \to 0,

where TM|_N denotes the restriction of the tangent bundle of M to N. This sequence arises from the differential of the inclusion map and reflects the local decomposition of tangent vectors along N into components tangent to N and normal to it. In the Grothendieck group of vector bundles, known as K-theory, the additivity of the group operation yields the relation [TN] + [\nu] = [TM|_N] for the classes of these virtual bundles.^[15] For an embedding of an n-dimensional manifold N into Euclidean space \mathbb{R}^N with N > n, the normal bundle \nu complements the tangent bundle TN in the trivial bundle \varepsilon^{N}|_N, so TN \oplus \nu \cong \varepsilon^N|_N. The stable normal bundle \nu^s, obtained by stabilizing \nu with trivial bundles if necessary, satisfies the same relation in the stable range. In reduced real K-theory \tilde{KO}(N), where the class of any trivial bundle vanishes, this implies [\nu^s] = -[TN], establishing the stable normal bundle as dual to the tangent bundle. This duality underpins applications in surgery theory and embedding obstructions.^[16] The Thom isomorphism relates the cohomology of the normal bundle to that of the submanifold. For an oriented vector bundle \nu of rank k over N, there exists a Thom class U \in H^k(\mathrm{Th}(\nu); \mathbb{Z}), where \mathrm{Th}(\nu) is the Thom space of \nu, inducing an isomorphism

H^*(N; \mathbb{Z}) \xrightarrow{\cup U} \tilde{H}^{*+k}(\mathrm{Th}(\nu); \mathbb{Z}).

This allows computation of the cohomology of the Thom space (and thus tubular neighborhoods) directly from the cohomology of N, with the inverse map given by the pushforward in cohomology. The isomorphism requires orientability of \nu, typically ensured when both N and M are orientable.^[16] The splitting TM|_N \cong TN \oplus \nu holds globally when M admits a Riemannian metric, as the normal bundle is then the orthogonal complement of TN in TM|_N, providing a canonical bundle projection that splits the exact sequence. Without a metric, the sequence may not split, but in the smooth category, metrics always exist, ensuring the isomorphism. Orientability conditions arise for preserving orientations: if M and N are orientable, then \nu is orientable (with first Stiefel-Whitney class w_1(\nu) = w_1(TM|_N) + w_1(TN) = 0), allowing an orientation-preserving splitting. Conversely, if \nu is non-orientable, no such compatible orientations exist on the summands.^[16]

Examples

Hypersurfaces

A hypersurface S in a manifold M of dimension m is a submanifold of codimension one, so \dim S = m-1. The normal bundle N_{M}S is then a real line bundle over S with one-dimensional fiber \mathbb{R}. The unit sphere S^{n-1} embedded in \mathbb{R}^n provides a concrete example of a hypersurface with trivial normal bundle. It admits a nowhere-vanishing global section given by the outward unit normal vector field \nu(p) = p for each p \in S^{n-1}, which trivializes N_{\mathbb{R}^n}S^{n-1}. For a hypersurface S in an orientable Riemannian manifold M, the normal bundle N_{M}S is orientable if and only if S is orientable, and since it is a real line bundle, this is equivalent to N_{M}S being trivial. In the orientable case, a consistent choice of unit normal vector field exists, providing a trivialization. For a non-orientable hypersurface, such as an immersion of the real projective plane \mathbb{RP}^2 into \mathbb{R}^3, the normal bundle is the non-trivial real line bundle over the base.^[17] The Gauss map \nu: S \to S^{m-1} assigns to each point of S its unit normal vector, viewed as an element of the unit sphere in the normal space within T_p M. The differential d\nu at a point equals the negative of the shape operator S_p: T_pS \to T_pS, which measures the extrinsic curvature. The determinant of the shape operator, \det S_p, equals the Gaussian curvature K_p at p \in S. The normal bundle N_{M}S is isomorphic to the dual of the determinant line bundle \det(TS).^[17]

Tori in Euclidean space

The 2-torus T^2 admits smooth embeddings in \mathbb{R}^3, such as the standard toroidal surface, for which the normal bundle is a rank-1 line bundle. However, a flat metric on T^2 cannot be realized isometrically in \mathbb{R}^3.^[18] To embed a flat torus smoothly without distorting the flat metric, one turns to the Clifford torus in \mathbb{R}^4, defined as the product of two circles of radius $1/\sqrt{2} lying on the unit sphere S^3 \subset \mathbb{R}^4.^[19] More generally, the Clifford torus T^k = (S^1)^k embeds in S^{2k-1} \subset \mathbb{R}^{2k} as the set of points (z_1, \dots, z_k) \in \mathbb{C}^k with |z_i| = 1/\sqrt{k} for each i. Since T^k is parallelizable (its tangent bundle is trivial) and the pullback of the trivial tangent bundle of \mathbb{R}^{2k} splits as the sum of the tangent and normal bundles, the normal bundle \nu of rank k is trivial. Moreover, as a minimal submanifold, the Clifford torus has a flat normal connection.^[20] In higher dimensions, consider an embedding of the n-torus T^n in \mathbb{R}^{2n}. The stable normal bundle \nu^{\text{st}} satisfies [\nu^{\text{st}}] = 0 in the real K-theory group \tilde{\text{KO}}(T^n), because T^n is stably parallelizable (its stable tangent bundle is trivial). A basic example is the embedding of T^1 = S^1 in \mathbb{R}^2 as the unit circle, where the normal bundle is the trivial line bundle.

Applications

Tubular neighborhoods

The tubular neighborhood theorem asserts that for a compact submanifold N of a smooth manifold M, there exists an open neighborhood U of N in M that is diffeomorphic to an open disk bundle D(\nu(N)) in the normal bundle \nu(N) of N in M, via a diffeomorphism that restricts to the inclusion of the zero section on N.^[21] This diffeomorphism provides a canonical model for the local geometry around N, allowing points in U to be uniquely identified with points in the normal bundle fibers. The theorem holds in the smooth category and extends to other settings like PL manifolds with appropriate modifications.^[21] In the case where M is equipped with a Riemannian metric, the diffeomorphism is constructed using the normal exponential map \exp^\perp: \nu(N) \to M, defined by \exp^\perp(p, v) = \exp_p(v) for p \in N and v \in N_p M normal to N at p, where \exp_p is the Riemannian exponential map at p. This map sends the zero section to N and, when restricted to a sufficiently small open disk bundle D_r(\nu(N)) of radius r > 0, provides the required diffeomorphism onto U = \exp^\perp(D_r(\nu(N))). The radius r is bounded above by the injectivity radius of the normal bundle, which ensures that normal geodesics do not intersect within this scale and that \exp^\perp is immersive with trivial kernel on the relevant domain. Tubular neighborhoods have key applications in transversality theory, where sections of the normal bundle can be used to perturb embeddings or immersions to achieve transverse intersections. Specifically, for a map f: P \to M and submanifold N \subset M, the tubular neighborhood U of N allows one to adjust f near f^{-1}(N) by adding a small section of the pullback normal bundle f^*\nu(N), ensuring transversality to N without altering the homotopy class of f. This perturbation technique underpins Thom's transversality theorem and facilitates the study of generic intersections in differential topology.^[21]

Symplectic manifolds

In symplectic geometry, consider a symplectic submanifold N of a symplectic manifold (M, \omega). The normal bundle \nu(N) of N in M inherits a natural fiberwise symplectic structure: at each point p \in N, the normal space \nu_p(N) is identified with the symplectic orthogonal complement (T_p N)^\omega = \{ v \in T_p M \mid \omega(v, w) = 0 \ \forall w \in T_p N \}, and the restriction of \omega to this space defines a non-degenerate symplectic form on the fibers.^[22] This symplectic normal bundle plays a central role in local models for embeddings and neighborhoods in symplectic manifolds.^[23] For the special case of a Lagrangian submanifold N, where \dim N = \frac{1}{2} \dim M and the pullback i^*\omega = 0 with i: N \hookrightarrow M the inclusion, the symplectic normal bundle \nu(N) is canonically symplectomorphic to the cotangent bundle T^*N equipped with its canonical symplectic form d\lambda, where \lambda is the Liouville 1-form.^[22] This identification arises from the fact that T_p M = T_p N \oplus (T_p N)^\omega decomposes symplectically, with (T_p N)^\omega \cong T_p^* N via the musical isomorphism induced by \omega. The Maslov class \mu(\nu(N)) \in H^1(N; \mathbb{Z}), an obstruction to a global trivialization of the complex structure on \nu(N) compatible with that on T^*N, measures the topological pairing between the two almost complex structures defining the symplectomorphisms along the fibers.^[24] This class captures essential features of the embedding, such as intersection obstructions in symplectic topology.^[25] Compatible tubular neighborhoods exist for such embeddings: by the Weinstein tubular neighborhood theorem, there is a neighborhood U of the zero section in T^*N and a symplectomorphism \psi: U \to V \subset M (with V a neighborhood of N) such that \psi preserves the symplectic forms and maps the zero section to N.^[22] This extends the classical tubular neighborhood theorem to the symplectic category, ensuring that local symplectic invariants are determined by the normal bundle data.^[23] In applications to symplectic topology, sections of the normal bundle \nu(N) correspond to Hamiltonian flows transverse to N. Specifically, a section s: N \to \nu(N) defines an embedding N \hookrightarrow M via the exponential map in the tubular neighborhood, and the Hamiltonian vector field generated by a primitive of the induced 1-form on the graph of s yields a flow that displaces N transversely, facilitating studies of displacement energy and spectral invariants.^[25]

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