Local diffeomorphism

In mathematics, particularly in differential geometry and topology, a local diffeomorphism is a smooth map f: M \to N between smooth manifolds M and N of the same dimension such that for every point x \in M, there exists a neighborhood U of x that f maps diffeomorphically onto an open subset f(U) of N.^[1] This local invertibility is equivalent to the differential df_x: T_x M \to T_{f(x)} N being a linear isomorphism at every point x.^[1] By the inverse function theorem, such maps are precisely those where the Jacobian matrix is invertible everywhere in local coordinates.^[2] Local diffeomorphisms are both smooth immersions and smooth submersions, ensuring they preserve the local structure of the manifolds without folding or collapsing dimensions.^[2] They form a fundamental concept in manifold theory, as compositions and finite products of local diffeomorphisms remain local diffeomorphisms, and a bijective local diffeomorphism is necessarily a global diffeomorphism.^[2] Unlike global diffeomorphisms, local diffeomorphisms need not be bijective; for instance, the exponential map \exp: \mathbb{R} \to S^1, given by t \mapsto (\cos t, \sin t), is a local diffeomorphism but covers the circle infinitely many times.^[1] These maps play a crucial role in applications such as covering spaces in algebraic topology, where universal covers are local diffeomorphisms,^[3] and in the study of Lie groups, where the exponential map provides a local diffeomorphism from the Lie algebra to the group.^[4] They also underpin change-of-variables formulas in multivariable calculus and enable the transport of geometric structures like metrics and tensors between manifolds.^[1]

Definition and Foundations

Formal Definition

A smooth manifold M is a second-countable Hausdorff topological space that is locally Euclidean, meaning every point has a neighborhood homeomorphic to an open subset of \mathbb{R}^n for some fixed n = \dim M, and it is equipped with a maximal atlas of charts whose transition maps are smooth (i.e., infinitely differentiable). Similarly, a smooth map f: M \to N between smooth manifolds M and N is a continuous map such that in local coordinates it is given by smooth functions. A diffeomorphism is a smooth bijective map with a smooth inverse, providing a notion of smooth equivalence between manifolds. Let M and N be smooth manifolds of the same dimension. A smooth map f: M \to N is a local diffeomorphism if for every point p \in M, there exists an open neighborhood U \subseteq M of p such that the restriction f|_U: U \to f(U) is a diffeomorphism onto its image. This condition implies that the differential df_p: T_p M \to T_{f(p)} N, the linear approximation of f at p between tangent spaces, is an isomorphism for each p \in M. The requirement that \dim M = \dim N is essential for local diffeomorphisms to exist non-trivially, as differing dimensions would prevent the differential from being invertible everywhere.

Equivalent Characterizations

A smooth map f: M \to N between smooth manifolds of the same dimension is a local diffeomorphism if and only if its differential df_p: T_p M \to T_{f(p)} N is a linear isomorphism for every p \in M.^[5] This condition ensures that f preserves the local structure of the manifolds by mapping tangent spaces invertibly at each point.^[6] Equivalently, f is a local diffeomorphism if and only if it is both a local immersion and a local submersion.^[7] A smooth map is a local immersion if df_p is injective for every p \in M, meaning it embeds tangent spaces without collapsing dimensions locally.^[1] A local submersion requires df_p to be surjective for every p \in M, ensuring the map covers the target tangent spaces fully in a neighborhood.^[8] When the manifolds have equal dimension, injectivity and surjectivity of the differential coincide with it being an isomorphism, yielding the equivalence.^[7] To see the equivalence between the invertible differential condition and local invertibility, consider the inverse function theorem for manifolds.^[5] Suppose df_p is an isomorphism at some p \in M. Choose smooth charts (U, \phi) around p in M and (V, \psi) around f(p) in N such that the coordinate representation of f, denoted F = \psi \circ f \circ \phi^{-1}, has invertible Jacobian matrix at \phi(p). By the classical inverse function theorem in \mathbb{R}^n, there exist neighborhoods W \subset \phi(U) and Z \subset \psi(V) where F is a diffeomorphism from W onto Z. Thus, f restricts to a diffeomorphism from \phi^{-1}(W) onto \psi^{-1}(Z), proving local invertibility.^[5] The converse follows because the differential of a local diffeomorphism must be invertible, as the chain rule applied to the local inverse yields the inverse linear map.^[6] This characterization also implies that f is locally a diffeomorphism onto its image: in a neighborhood of each p, f maps diffeomorphically onto an open submanifold of N contained in the image of f.^[5] The inverse function theorem application in the manifold setting, via compatible charts reducing to the Euclidean case, directly establishes this local bijectivity onto the image while preserving smoothness.^[8]

Core Properties

Invertibility Conditions

A local diffeomorphism f: M \to N between smooth manifolds of the same dimension ensures the existence of a local inverse at every point p \in M. Specifically, there exists an open neighborhood U \subset M of p and an open neighborhood V = f(U) \subset N of f(p) such that f|_U: U \to V is a diffeomorphism, with a smooth inverse g: V \to U satisfying g \circ f = \mathrm{id}_U and f \circ g = \mathrm{id}_V. This local invertibility follows directly from the inverse function theorem, which guarantees that the map is smoothly invertible on sufficiently small neighborhoods when the differential df_p: T_p M \to T_{f(p)} N is an isomorphism. The construction of the local inverse g relies on the invertibility of the differential. At the point f(p), the differential of the inverse satisfies dg_{f(p)} = (df_p)^{-1}, the linear isomorphism inverse to df_p. This linear approximation is extended to a smooth local diffeomorphism g on a neighborhood V of f(p) via the inverse function theorem, ensuring that g smoothly inverts f locally while preserving the manifold structure.^[9] Although local diffeomorphisms are invertible locally, they may fail to be global diffeomorphisms without additional conditions. For instance, if M is compact and f: M \to N is a bijective local diffeomorphism between manifolds of the same dimension, then f is a global diffeomorphism, as bijectivity combined with compactness implies f is a homeomorphism, and local smoothness extends globally. Similarly, for maps between non-compact manifolds like \mathbb{R}^n, properness of f (meaning preimages of compact sets are compact) together with local invertibility everywhere ensures f is a global diffeomorphism onto its image. A counterexample illustrating failure of global invertibility is the map f: S^1 \to S^1 given by f(e^{i\theta}) = e^{2i\theta}, which is a local diffeomorphism but not injective globally.^[9] Local diffeomorphisms also play a key role in covering maps within differential geometry. A smooth covering map is a local diffeomorphism that is a covering projection in the topological sense, meaning every point in N has a neighborhood evenly covered by f. If the base manifold N is simply connected, such a covering map is a global diffeomorphism. In this context, local diffeomorphisms correspond to étale maps when viewed through the lens of smooth infinity-groupoids or synthetic differential geometry, where étaleness signifies formal local invertibility in the differential structure.^[10]

Topological Implications

A local diffeomorphism f: M \to N between smooth manifolds is an open map, meaning that for every open set U \subset M, the image f(U) is open in N. This property arises because f is a local homeomorphism: by the inverse function theorem, around each point p \in M, there exist neighborhoods V \subset M of p and W \subset N of f(p) such that f|_V: V \to W is a homeomorphism. To see why local homeomorphisms are open, consider an arbitrary open set U \subset M. Cover U by such neighborhoods V_i where each f(V_i) is open in N. Then f(U) = \bigcup f(V_i \cap U) is a union of open sets, hence open. The smoothness of f ensures that this topological openness is compatible with the manifold structures.^[11]^[5] Under suitable global conditions, local diffeomorphisms exhibit behavior akin to covering maps. In particular, if f: M \to N is a proper local diffeomorphism between connected smooth manifolds (meaning preimages of compact sets are compact), then f is a smooth covering map. This result, known as Proposition 4.46 in John M. Lee's Introduction to Smooth Manifolds, follows from the local homeomorphism property combined with properness, which ensures evenly covered neighborhoods and surjectivity without pathological branching. For instance, if the domain M is compact, properness holds automatically, making compact-to-any local diffeomorphisms covering maps. Such coverings play a key role in understanding global topology through local smooth structure.^[5] Local diffeomorphisms also have significant implications for homotopy and orientability. Being local homeomorphisms, they induce isomorphisms on the fundamental groups of sufficiently small neighborhoods, where these groups are typically trivial (as small balls in manifolds are simply connected), thus preserving local homotopy types. More globally, when f: M \to N is a covering map arising from a local diffeomorphism, it induces an injective homomorphism on fundamental groups, reflecting the deck transformation action. Regarding orientability, if N is an orientable manifold and f: M \to N is a local diffeomorphism with M connected, then M is also orientable. This holds because the local inverses of f allow the orientation on N to be consistently pulled back to M, either preserving or reversing it uniformly across the manifold, enabling a global orientation on M.^[5]^[12]

Illustrative Examples

Positive Examples

One prominent example of a local diffeomorphism arises in the context of Lie groups, where the exponential map \exp: \mathfrak{g} \to G from the Lie algebra \mathfrak{g} to the Lie group G serves as a local diffeomorphism near the origin $0 \in \mathfrak{g}. The differential at the origin, d\exp_0: T_0 \mathfrak{g} \to T_e G, is the identity map on \mathfrak{g} (identifying tangent spaces appropriately), which is clearly invertible, ensuring that \exp is locally invertible with a smooth inverse in a neighborhood of $0. This property holds for any Lie group and is fundamental for constructing one-parameter subgroups and understanding local structure around the identity.^[5] Projection maps provide another illustrative case, particularly when viewed through the lens of submersions between manifolds of the same dimension. For instance, the projection \pi: M \times N \to M is a smooth submersion, with its differential d\pi_{(p,q)}: T_{(p,q)}(M \times N) \to T_p M being surjective (projecting onto the M-component while ignoring the N-component), and it becomes a local diffeomorphism precisely when \dim N = 0 (i.e., N is a point), in which case d\pi is an isomorphism. More generally, submersions between equal-dimensional manifolds are local diffeomorphisms, as the surjective differential of maximal rank implies local invertibility.^[5] In the construction of smooth manifolds, transition maps between compatible charts in an atlas exemplify local diffeomorphisms by design. For charts (U, \phi) and (V, \psi) overlapping on M, the transition map \psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V) is a diffeomorphism between open subsets of \mathbb{R}^n, with its differential being an invertible linear map at every point due to the smoothness and bijectivity requirements of the atlas. This ensures the manifold's differential structure is well-defined and independent of chart choice, verifying local diffeomorphism status through the existence of smooth inverses.^[5] Smooth covering maps offer a classical geometric example, such as the universal covering \pi: \mathbb{R} \to S^1 given by \pi(t) = e^{2\pi i t}. This map is a local diffeomorphism because, at any t \in \mathbb{R}, the differential d\pi_t: T_t \mathbb{R} \to T_{\pi(t)} S^1 (identifying T \mathbb{R} \cong \mathbb{R} and T S^1 \cong \mathbb{R}) is multiplication by $2\pi, which is invertible (nonzero scalar). Locally, around any point on S^1, the preimage intervals in \mathbb{R} map diffeomorphically, illustrating how covering maps between same-dimensional manifolds inherit local diffeomorphism properties.^[5]

Counterexamples

A prominent counterexample illustrating the failure of a map to be a local diffeomorphism arises from dimension mismatch in the inclusion of submanifolds. Consider the standard embedding i: S^1 \to \mathbb{R}^2 defined by i(\theta) = (\cos \theta, \sin \theta) for \theta \in [0, 2\pi). This map is an immersion because its differential di_\theta: T_\theta S^1 \to T_{i(\theta)} \mathbb{R}^2 at each point is injective: identifying T_\theta S^1 \cong \mathbb{R} with the basis vector \frac{d}{d\theta}, we have di_\theta \left( \frac{d}{d\theta} \right) = (-\sin \theta, \cos \theta), which is nonzero and thus spans a 1-dimensional subspace of \mathbb{R}^2. However, it is not a local diffeomorphism since di_\theta is not surjective (the image is a line, not the full plane), violating the requirement that the differential be an isomorphism everywhere.^[13] Constant maps provide another trivial failure mode. For instance, the constant map c: \mathbb{R}^n \to \mathbb{R}^m with c(p) = q for fixed q has differential dc_p = 0 at every point p, which is neither injective nor surjective unless n = m = 0. Thus, it cannot be a local diffeomorphism, as the zero map is not an isomorphism between tangent spaces.^[13] In the case of same-dimensional manifolds, non-invertible differentials at certain points prevent local diffeomorphism status. A classic example is the smooth map f: \mathbb{R}^2 \to \mathbb{R}^2 given by f(x,y) = (x, xy). The Jacobian matrix at a point (x,y) is

\begin{pmatrix} 1 & 0 \\ y & x \end{pmatrix},

with determinant x. This determinant vanishes along the line x = 0, where the rank drops to 1, making df_{(0,y)} non-invertible (its image is the line spanned by (1, y)^T in \mathbb{R}^2). Consequently, f fails to be a local diffeomorphism at points on this line, as the inverse function theorem does not apply there.^[14] While global diffeomorphisms, such as \tan: (-\pi/2, \pi/2) \to \mathbb{R}, are necessarily local diffeomorphisms (with differential d(\tan)_x = \sec^2 x > 0, hence invertible), the converse does not hold; non-proper maps like the universal covering map \pi: \mathbb{R}^2 \to [T^2](/page/T+2), (x,y) \mapsto (e^{2\pi i x}, e^{2\pi i y}), are local diffeomorphisms (differential is the identity isomorphism) but not global ones due to lack of bijectivity. However, the above examples demonstrate breakdowns in the local condition itself.^[13]

Applications in Geometry and Dynamics

Role in Manifold Theory

Local diffeomorphisms play a fundamental role in defining the smooth structure of manifolds through the compatibility of coordinate charts in an atlas. A smooth manifold is equipped with an atlas consisting of charts (U_i, \phi_i), where each \phi_i: U_i \to \mathbb{R}^n is a homeomorphism onto an open set, and the transition maps \phi_j \circ \phi_i^{-1} on overlaps U_i \cap U_j are diffeomorphisms, ensuring that the manifold's smoothness is independent of the choice of coordinates.^[15] This compatibility condition relies on local diffeomorphisms to glue local Euclidean charts into a global smooth structure, preserving differentiability across the manifold.^[16] The invariance of the smooth structure under changes of coordinates is guaranteed by the diffeomorphic nature of transition maps, which allow equivalent atlases to define the same manifold up to diffeomorphism. Two atlases are equivalent if the identity map between them is a diffeomorphism, meaning that any refinement or compatible extension yields the same differentiable category.^[17] In quotient manifolds, the submersion theorem ensures that if a Lie group acts freely and properly on a manifold M, the projection \pi: M \to M/G is a smooth submersion, locally equivalent to a product of a diffeomorphism and a projection, thereby inducing a smooth structure on the orbit space M/G.^[18] Historically, local diffeomorphisms underpin results like Ehresmann's fibration theorem, which states that a proper submersion between smooth manifolds is a locally trivial fibration, with fibers diffeomorphic via local diffeomorphic trivializations.^[19] This theorem highlights how submersions, as local surjections with surjective differentials, facilitate the study of fibrations where fiber bundles are locally products involving diffeomorphic slices. Every smooth manifold admits the identity map as a local diffeomorphism to itself, serving as the canonical example of a bijective smooth map with smooth inverse.^[16]

Local Flows and Diffeomorphisms

In dynamical systems on smooth manifolds, a flow generated by a smooth vector field X on a manifold M is a smooth map \phi: D \to M, where D \subseteq \mathbb{R} \times M is an open subset containing \{0\} \times M, satisfying \phi(0, p) = p for all p \in M and the flow property \phi(t + s, p) = \phi(t, \phi(s, p)) whenever both sides are defined in D. This construction arises from solving the autonomous ordinary differential equation \frac{d}{dt} \gamma(t) = X(\gamma(t)) with initial condition \gamma(0) = p, yielding integral curves \gamma_p(t) = \phi(t, p) that parametrize the trajectories of X. The existence and uniqueness of such maximal integral curves, as well as the smoothness of \phi, follow from standard theorems on ODEs on manifolds. For small |t|, each time slice \phi_t: \dom(\phi_t) \to M, where \dom(\phi_t) = \{p \in M \mid (t, p) \in D\}, is a local diffeomorphism whenever X is smooth. This property holds because the differential d\phi_t at any point satisfies the linearized variation equation along the integral curve, a linear ODE with initial condition the identity map at t = 0, which remains invertible for sufficiently small t by the time-dependent inverse function theorem. Equivalently, the fact that the Lie derivative \mathcal{L}_X X = 0 ensures the flow preserves the generating vector field, and continuity of d\phi_t from the identity implies local invertibility of the differential, consistent with characterizations of local diffeomorphisms via invertible differentials.^[20] For example, on \mathbb{R} with a constant nonzero vector field, the flow consists of translations along the real line, which are global diffeomorphisms and illustrate how flows generate local coordinate changes. A key application arises in contact geometry, where the Reeb vector field R_\alpha on a contact manifold (M, \alpha)—defined by \alpha(R_\alpha) = 1 and i_{R_\alpha} d\alpha = 0—generates a flow consisting of contact diffeomorphisms, which are local diffeomorphisms preserving the contact structure \xi = \ker \alpha. These Reeb flows play a central role in studying periodic orbits and dynamics on contact manifolds, such as tight contact structures on three-manifolds. If X is complete, meaning D = \mathbb{R} \times M, the flow extends to a global one-parameter group \{\phi_t\}_{t \in \mathbb{R}} of diffeomorphisms on M, forming a \mathbb{R}-action by local diffeomorphisms that satisfy the group axioms under composition.^[20] Vector fields with compact support are always complete, ensuring such global extensions in many geometric contexts.