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Submanifold

In , a submanifold (typically meaning an submanifold) is a N of a smooth manifold M that is itself a smooth manifold of lower , inheriting the and from M, such that for every point p \in N, there exists a (U, \phi) of M with p \in U where \phi(N \cap U) is an open subset of a of the model . This local Euclidean-like appearance ensures that N behaves as a manifold of lower within the ambient M. Submanifolds are classified into and immersed types, with embedded submanifolds requiring the i: N \to M to be a onto its image, preserving the , while immersed submanifolds arise from immersions (smooth maps with everywhere injective differentials), where the image may self-intersect if the map is not injective or fail to inherit the subspace topology. Embedded submanifolds are diffeomorphic to their images via proper embeddings, ensuring clean topological integration, whereas immersed ones allow for more flexible but potentially singular global embeddings, as studied in theorems like Whitney's embedding theorem. Classic examples include the unit sphere S^{m-1} = \{ x \in \mathbb{R}^m \mid \|x\| = 1 \}, which is an embedded submanifold of codimension 1 in \mathbb{R}^m, and graphs of smooth functions y = f(x), forming 1-dimensional submanifolds in \mathbb{R}^2. The 2-torus T^2 = S^1 \times S^1 serves as an embedded submanifold in \mathbb{R}^4 or \mathbb{R}^3 under standard embeddings. These structures are fundamental in differential geometry, enabling the analysis of tangent spaces as subspaces of the ambient tangent space, level sets of smooth functions with non-vanishing gradients, and applications in Riemannian geometry such as totally geodesic submanifolds and curvature inheritance.

Introduction

Overview

In , a submanifold is a of a smooth manifold that inherits a smooth manifold structure from its ambient space, allowing it to be treated as a manifold in its own right while preserving local compatibility with the surrounding geometry. This concept generalizes the idea of lower-dimensional objects, such as curves or surfaces, within higher-dimensional spaces, providing a framework for analyzing their intrinsic properties relative to the larger manifold. Not every of a manifold qualifies as a submanifold; the key requirement is that the from the subset to the ambient manifold must be smooth, ensuring that around each point, the subset locally resembles a flat in appropriate coordinates. This compatibility with the ambient manifold's atlas distinguishes submanifolds from mere topological subsets, enabling the transfer of structures like spaces and metrics. Submanifolds are essential for modeling lower-dimensional phenomena in higher-dimensional contexts, such as particle trajectories in spaces of systems or surfaces in optimization problems. They facilitate the restriction of geometric tools from the ambient manifold, supporting applications in physics, theory, and the study of partial equations. A brief distinction exists between immersed submanifolds, defined via smooth immersions that may allow self-intersections, and submanifolds, which are proper subsets without such global overlaps.

Historical Context

The concept of submanifolds originated in the early with Carl Friedrich Gauss's foundational work on curved surfaces in three-dimensional . In his 1827 paper Disquisitiones Generales Circa Superficies Curvas, Gauss introduced the distinction between intrinsic geometry, measurable solely on the surface itself, and extrinsic geometry, dependent on the in ambient . This laid the groundwork for understanding submanifolds as lower-dimensional objects within higher-dimensional spaces, emphasizing quantities like that are independent of the . During the mid-19th century, and Gaston Darboux extended these ideas to higher dimensions, paving the way for abstract analogs of surfaces. , Über die Hypothesen, welche der Geometrie zu Grunde liegen, generalized Gauss's surface theory to n-dimensional manifolds equipped with a , introducing and the notion of manifolds as locally spaces that could serve as models for submanifolds in broader contexts. Darboux, building on this in the late 19th century, developed tools for studying hypersurfaces and integral manifolds in higher-dimensional spaces, including theorems on the local structure of submanifolds defined by partial equations. The early 20th century saw formalization through Élie Cartan's generalizations and Hassler Whitney's embedding results. In the 1930s, Cartan advanced the theory of abstract manifolds using moving frames and affine connections, allowing submanifolds to be studied independently of specific embeddings in and integrating actions into . Whitney's 1936 theorems distinguished immersions—local embeddings without self-intersections—from global embeddings, proving that any smooth n-dimensional manifold immerses in \mathbb{R}^{2n} and embeds in \mathbb{R}^{2n+1}, thus clarifying the topological conditions for submanifolds in . Post-1950s developments shifted submanifold theory from Euclidean-specific constructions to general smooth manifolds, influenced by advances in and . John Milnor's theorem in the 1960s, building on earlier work, provided tools for classifying smooth structures on manifolds, enabling the study of exotic spheres and classifications. This era marked a broader integration with global , emphasizing abstract properties over concrete embeddings.

Definitions

Immersed Submanifolds

An immersed submanifold of dimension k in an n-dimensional manifold N (with k \leq n) is defined as a pair (M, f), where M is a k-manifold and f: M \to N is a immersion. A map f: M \to N is an immersion if its df_p: T_p M \to T_{f(p)} N is injective for every point p \in M. This condition ensures that the rank of the is constantly k everywhere: \rank(df_p) = \dim(M) = k \quad \forall \, p \in M. The image f(M) inherits a manifold structure from M via f, making it an immersed submanifold of N, though it may not be closed or properly embedded in N. Locally, every immersed submanifold behaves like an embedded one. Specifically, for each point q \in f(M), there exist coordinate charts around a preimage point in M and around q in N such that the immersion f appears as the inclusion of an open subset of \mathbb{R}^k \times \{0\}^{n-k} into \mathbb{R}^n, or equivalently, as a graph over a k-dimensional coordinate subspace in the chart on N. This local embedding property highlights that immersions preserve the differential structure near each point without requiring global injectivity. A example is the figure-eight in \mathbb{R}^2, obtained as the of the f: \mathbb{R} \to \mathbb{R}^2 given by f(t) = (\sin t, \sin 2t). This parametrizes a 1-manifold locally diffeomorphic to \mathbb{R}, but the map is not injective since f(t + 2\pi k) = f(t) for any integer k, causing the to self-intersect at the (for instance, f(0) = f(\pi) = (0,0)), forming a that is immersed yet not embedded globally. A key result is that every is locally an : for any point p \in M, there is a neighborhood U \subset M such that f|_U: U \to N is a onto its , equipped with the . This underscores the local Euclidean nature of immersed submanifolds, distinguishing them from embeddings primarily by allowing global self-intersections.

Embedded Submanifolds

An embedded submanifold arises from a , which is a f: M \to N between manifolds that is both an and a topological . Specifically, f is an if its df_p: T_p M \to T_{f(p)} N is injective for every p \in M, ensuring local properties, and it is a topological if f is a onto its f(M) equipped with the induced from N. This means f is continuous, injective, and its inverse f^{-1}: f(M) \to M is continuous with respect to the on f(M). The f(M) then inherits a manifold structure from N, where charts on N restrict to charts on f(M) that form a atlas compatible with the original structure on M. The requirement that f be a homeomorphism onto its image distinguishes embedded submanifolds from mere immersed submanifolds by enforcing global topological well-behavedness. In an immersion without the embedding condition, the image may exhibit pathologies such as self-intersections or dense windings, where the map is locally injective but globally non-injective, leading to a subspace topology that does not align with the manifold topology on M. For embedded submanifolds, the image f(M) is properly embedded in N, meaning it is a closed subset if M is compact, and the inclusion map induces a diffeomorphism between M and f(M), preserving all local and global properties without overlaps. This global injectivity ensures that the subspace topology on f(M) coincides exactly with the topology transported from M via f. A cornerstone result guaranteeing the existence of such embeddings is the , which asserts that every k-dimensional manifold M (Hausdorff and second-countable) admits a into \mathbb{R}^{2k}. For compact manifolds, this embedding is closed, meaning the image is a compact subset of \mathbb{R}^{2k}, and the is induced from the . Proved by Hassler , this theorem highlights the interplay between local properties and global , allowing any manifold to be realized as an embedded submanifold of without self-intersections. The proof involves approximating continuous maps with ones and resolving intersections using arguments in sufficiently high dimensions.

Variations

Open submanifolds arise as open subsets of a smooth manifold M, inheriting the from M such that the i: U \to M is a embedding. This construction ensures that U is a submanifold of the same as M, with charts restricted from those of M, preserving the and . Such submanifolds are particularly useful in local analysis, as they allow seamless extension of functions and vector fields from the subset to neighborhoods in the ambient manifold. Closed submanifolds, often compact without boundary, refer to submanifolds that are closed subsets of the ambient manifold M. Compactness in this context implies that the submanifold is proper as an , ensuring global topological control, as seen in examples like real projective spaces in spheres. These structures are stable under compact perturbations and play a key role in theorems on fixed points and index theory. Weak submanifolds extend the concept to settings where the differential is injective , rather than everywhere, which is essential in for handling variational problems with singularities. A weak immersion \tilde{\Phi}: \Sigma \to \mathbb{R}^m of a surface \Sigma is a Lipschitz map that is weakly conformal and has an L^2-bounded Gauss map, allowing for in energy spaces like the Willmore functional. This notion captures integral currents and varifolds as limits of immersions, facilitating the study of minimal surfaces beyond classical . Hypersurfaces represent codimension-1 submanifolds, typically realized as level sets g^{-1}(c) of a g: N \to \mathbb{R} where the \nabla g never vanishes on the level set. This construction yields an submanifold diffeomorphic to the zero set of a defining with non-vanishing , enabling the use of coarea formulas and vector fields. Hypersurfaces are fundamental in separating ambient spaces and appear in and boundary value problems. In semi-Riemannian geometry, almost submanifolds generalize submanifolds by allowing the second fundamental form to satisfy a control condition rather than vanishing identically, accommodating indefinite metrics. These structures arise from second-order on the manifold, where the curvature tensor aligns with a of subspaces, providing a for pseudo-Riemannian analogs of classical geometry. Singular submanifolds, or singular varieties in , feature points where the local ring is not regular, meaning the dimension of the exceeds the dimension of the , leading to non-smooth behavior. These extend smooth submanifolds by including loci defined by polynomial equations with critical points, bridging differential and algebraic geometry through .

Properties

Local Properties

A submanifold inherits its differentiable structure from the ambient manifold through an induced atlas, where local charts are constructed such that the submanifold appears as a over a coordinate . Specifically, for an immersed submanifold S \subset N given by a map f: M \to N, around each point p \in M, there exists a (U, \phi) on N such that f(U) is graphed over \phi(U) \cong \mathbb{R}^k \times \mathbb{R}^l with k = \dim M, and the induced chart on S is the restriction ensuring compatibility via the . This construction ensures that the transition maps between charts on S are , as they compose with those of N. The tangent space to the submanifold at a point q = f(p) \in S is defined as the image of the differential, T_q S = df_p(T_p M), which forms a subspace of the tangent space T_q N of the ambient manifold. In the case where N is equipped with a Riemannian metric, the normal space N_q S is the orthogonal complement to T_q S in T_q N with respect to this metric, providing a direct sum decomposition T_q N = T_q S \oplus N_q S. This subspace structure captures the directions tangent to S and allows for the projection of vectors from N onto S. When the ambient manifold N is Riemannian with metric g_N, the submanifold S inherits a Riemannian metric g_S = f^* g_N, the pullback of g_N via the immersion f, which restricts to an inner product on each tangent space T_q S. This induced metric makes S a Riemannian submanifold, enabling the measurement of lengths and angles intrinsically on S while reflecting the embedding in N. Locally, in coordinates where f(u) = (u, 0) near the origin, the components of g_S are g_{ij}(u) = \langle \partial f / \partial u^i, \partial f / \partial u^j \rangle_{g_N}, forming a positive-definite matrix. The second fundamental form quantifies the extrinsic of the submanifold, measuring how geodesics on S deviate from those in N. It is defined as a symmetric bilinear tensor \mathrm{II}: T_q S \times T_q S \to N_q S, given by the normal component of the : \mathrm{II}(X, Y) = (\nabla_X^{\, N} Y)^\perp, where \nabla^N is the connection on N and ^\perp denotes projection onto bundle. The symmetry \mathrm{II}(X, Y) = \mathrm{II}(Y, X) follows from the torsion-freeness of \nabla^N and the compatibility, ensuring \mathrm{II} is a well-defined (0,2)-tensor when paired with the normal space. For hypersurfaces, it reduces to a scalar-valued form via a unit \nu, \mathrm{II}(X, Y) = \langle \nabla_X^{\, N} Y, \nu \rangle \nu. This form encodes the bending of S in N, independent of the choice of up to sign. The Gauss equation relates the intrinsic Riemann curvature tensor R^S on S to that of N and the second fundamental form, highlighting how extrinsic geometry influences intrinsic properties. Precisely, for tangent vectors X, Y, Z \in T_q S, R^S(X, Y) Z = \mathrm{proj}_{T_q S} \left( R^N(X, Y) Z \right) + \mathrm{II}(X, Z) \cdot Y - \mathrm{II}(Y, Z) \cdot X, where the projection is onto the tangent space and the wedge product-like term arises from the commutator of \mathrm{II}. In index-free form, the curvature operator satisfies R^S = \mathrm{proj}_{T} (R^N) + [\mathrm{II}, \mathrm{II}], with the commutator capturing the interaction between normal directions. For the scalar curvature or sectional curvatures, this implies that the Gaussian curvature of S (in codimension 1) is the product of principal curvatures adjusted by the ambient curvature. This equation, derived from the definition of the induced connection on S, \nabla_X^S Y = (\nabla_X^N Y)^T, underscores the local embedding's effect on geodesics and volumes. Associated with the second fundamental form is the Weingarten map, a family of endomorphisms on the tangent space parameterized by the normal space, describing the variation of the normal bundle along tangent directions. For a normal vector field \nu \in N_q S, the Weingarten map A_\nu: T_q S \to T_q S is defined by A_\nu(X) = - (\nabla_X^N \nu)^T, the negative tangent component of the ambient covariant derivative. It satisfies the duality g_S(A_\nu(X), Y) = g_N(\mathrm{II}(X, Y), \nu), making A_\nu self-adjoint with respect to g_S, and its eigenvalues are the principal curvatures in the direction of \nu. In coordinates, if \nu is a unit normal, the matrix of A_\nu is [A_i^j] = g^{jk} L_{ki}, where L_{ij} = g_N(\mathrm{II}(\partial_i, \partial_j), \nu). This map provides a linear algebraic tool for analyzing local rigidity and stability of the submanifold.

Global Properties

A closed embedded submanifold of a compact manifold without boundary is itself compact. This follows from the fact that such a submanifold is a closed in the of the ambient compact manifold, and closed subsets of compact Hausdorff spaces are compact. Compactness implies that the submanifold is bounded in any into , ensuring that its image under a continuous remains contained within a finite cover of coordinate charts. Orientability of a submanifold is not necessarily inherited from an orientable ambient manifold; for example, the is a non-orientable submanifold of ℝ³. However, codimension-zero submanifolds of an orientable manifold are orientable, with orientations corresponding via determinants in compatible charts. The tube lemma ensures that neighborhoods around the submanifold allow consistent orientation choices aligned with the ambient , facilitating the extension of orientations locally when applicable. The induced by a submanifold \iota: S \hookrightarrow M pulls back differential forms, yielding a map on groups \iota^*: H^*_{dR}(M) \to H^*_{dR}(S). For a closed oriented submanifold S of k in an oriented manifold M of m, this inclusion defines the Poincaré dual Pd_M(S) \in H^{m-k}_{dR}(M), such that for any closed (m-k)-form \omega on M, the over S of the pullback of a compactly supported k-form equals the pairing via \omega. Poincaré duality for closed oriented submanifolds establishes an isomorphism H^k_{dR}(M) \cong (H^{m-k}_c(M))^*, where the dual element corresponds to integration over transversal intersections with complementary-al submanifolds. This duality links homology classes represented by submanifolds to cohomology via intersection pairings, with the sign determined by orientation agreement at intersection points. In high codimension, submanifolds of positive greater than zero have measure zero with respect to the on the ambient or product measures on manifolds. This rarity arises because such submanifolds can be covered by countably many graphs over lower-dimensional slices, each of which has measure zero by Fubini's theorem, and countable unions preserve measure zero. The transversality theorem ensures that generic intersections of submanifolds are themselves submanifolds of expected , provided the span the ambient tangent space at intersection points; specifically, for submanifolds X_1, X_2 \subset M, if T_p X_1 + T_p X_2 = T_p M for p \in X_1 \cap X_2, then X_1 \cap X_2 is a submanifold of \dim X_1 + \dim X_2 - \dim M. Nash's embedding theorem states that any compact smooth of dimension n can be isometrically embedded into \mathbb{R}^N for sufficiently large N, specifically N = \frac{n(3n+11)}{2} as per Nash's 1956 result. This result, building on earlier work for C^1 embeddings, guarantees global realization of the while preserving the manifold's intrinsic . For immersed submanifolds, the of the map, defined via integration of forms, provides a global invariant measuring the oriented covering multiplicity.

Examples in Euclidean Space

Low-Dimensional Cases

In low-dimensional Euclidean spaces, submanifolds of dimension 1 manifest as curves, while those of dimension 2 appear as surfaces. A curve in \mathbb{R}^2 or \mathbb{R}^3 is typically parametrized by a smooth map \gamma: I \to \mathbb{R}^n, where I \subset \mathbb{R} is an interval, and for regularity, the derivative \gamma' is nowhere zero. To study intrinsic properties like curvature, curves are often reparametrized by arc length s, yielding a unit-speed parametrization where |\gamma'(s)| = 1. For such unit-speed curves in \mathbb{R}^3, the Frenet frame provides an adapted consisting of the T(s) = \gamma'(s), the principal N(s), and the binormal B(s) = T(s) \times N(s). The \kappa(s) measures how much the curve bends away from the tangent, given by \kappa(s) = |\gamma''(s)| when parametrized by , while the torsion \tau(s) quantifies twisting out of the spanned by T and N. The fundamental theorem of space curves asserts that given continuous functions \kappa(s) > 0 and \tau(s) on an interval, there exists a unique (up to rigid motion) unit-speed curve \gamma(s) in \mathbb{R}^3 with those curvature and torsion functions. This theorem underscores how \kappa and \tau fully determine the local of the curve. A classic example of an 1-dimensional submanifold is S^1, realized in \mathbb{R}^2 via the parametrization \gamma(\theta) = (r \cos \theta, r \sin \theta) for \theta \in [0, 2\pi), which is smooth and injectively maps onto its image without self-intersections. In contrast, the helix \gamma(t) = (a \cos t, a \sin t, b t) for t \in \mathbb{R} and constants a, b > 0 provides an embedded curve in \mathbb{R}^3, as its parametrization is a smooth embedding that winds indefinitely without self-intersections, yielding constant curvature \kappa = a/(a^2 + b^2) and torsion \tau = b/(a^2 + b^2). Turning to 2-dimensional submanifolds, surfaces in \mathbb{R}^3 can be represented as graphs over a domain in \mathbb{R}^2, such as z = f(x,y) where f is , ensuring the map (x,y) \mapsto (x, y, f(x,y)) is an . Alternatively, parametric representations use X: U \to \mathbb{R}^3 with U \subset \mathbb{R}^2 open and X_u \times X_v \neq 0, providing a local coordinate chart. For these surfaces, the K is the product of the principal curvatures \kappa_1 \kappa_2, capturing intrinsic bending, while the H is the trace of the second fundamental form, averaging the principal curvatures; surfaces with H = 0 everywhere are minimal and minimize area among nearby variations.

Higher-Dimensional Constructions

One fundamental method for constructing submanifolds of dimension k \geq 3 in \mathbb{R}^n (with n > k) involves regular level sets of smooth functions. Consider a smooth map F: \mathbb{R}^n \to \mathbb{R}^{n-k} such that c \in \mathbb{R}^{n-k} is a regular value, meaning the differential dF_p has full rank n-k at every point p in the preimage S = \{ x \in \mathbb{R}^n \mid F(x) = c \}. Then S is a smooth embedded submanifold of \mathbb{R}^n with dimension k. For a hypersurface (codimension 1, so k = n-1), where F: \mathbb{R}^n \to \mathbb{R} and \nabla F \neq 0 on S, the unit normal vector is given by \nu = \nabla F / |\nabla F|. Sard's theorem guarantees the existence of such regular values for generic choices of c. Specifically, for a smooth map F: \mathbb{R}^n \to \mathbb{R}^m with m \leq n, the set of critical values has zero in \mathbb{R}^m, ensuring that almost every c yields a regular level set that is a submanifold of codimension m. This applies even when the domain is a higher-dimensional manifold, but for ambient space, it provides a robust tool for constructing submanifolds of arbitrary dimension k \geq 3 by selecting m = n - k. Another construction uses graphs of smooth functions. For a smooth map f: \mathbb{R}^k \to \mathbb{R}^{n-k}, the graph S = \{ (x, f(x)) \mid x \in \mathbb{R}^k \} is a smooth embedded submanifold of \mathbb{R}^n diffeomorphic to \mathbb{R}^k. Locally, any embedded submanifold admits such a graph representation near each point, by the implicit function theorem: if S is defined locally by F(y, z) = 0 with y \in \mathbb{R}^k, z \in \mathbb{R}^{n-k}, and the partial derivative \partial F / \partial z is invertible, then z = g(y) solves the equation smoothly near the point. This yields open submanifolds but can be compactified for closed examples in higher dimensions. Product constructions extend lower-dimensional embeddings to higher dimensions. The product S^m \times S^l (dimension m + l \geq 3) embeds smoothly into \mathbb{R}^{m + l + 2} via the standard inclusions S^m \hookrightarrow \mathbb{R}^{m+1} and S^l \hookrightarrow \mathbb{R}^{l+1}, taking the product in \mathbb{R}^{m+1} \times \mathbb{R}^{l+1}. For instance, the S^3 embeds in \mathbb{R}^4 as the \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1 \}, a regular level set of F(x) = \|x\|^2 - 1. Similarly, the 2-torus T^2 = S^1 \times S^1 embeds in \mathbb{R}^3 via the standard parametrization (u, v) \mapsto ( (R + r \cos v) \cos u, (R + r \cos v) \sin u, r \sin v ) for R > r > 0, which produces an surface without self-intersections.