Manifold - Maps and Functions

Maps Between Manifolds and Functions on Manifolds Maps Between Manifolds Continuous and Smooth Maps When studying manifolds, we need to understand what kinds of maps between them preserve the geometric structure we care about. A continuous map between two topological manifolds is simply a map that preserves the topological structure—roughly speaking, nearby points map to nearby points. However, continuity alone tells us nothing about whether the map respects the smooth structure that makes manifolds interesting. A smooth map (also called a $C^\infty$ map) is much more restrictive: it respects the smooth structure of both manifolds. More precisely, if $f: M \to N$ is a smooth map between manifolds with coordinate systems, then when we express $f$ in local coordinates, we get functions that are infinitely differentiable. This means we can compute derivatives of $f$ to any order we like. The key insight is that smoothness is a local property—we only need to verify it in coordinate neighborhoods. Once you know $f$ is smooth in local coordinates around each point, you're done. Why does this matter? Continuous maps are too general; smooth maps allow us to use calculus, which opens up all the tools of differential geometry. Immersions and Submersions Once we have smooth maps, the next natural question is: what can we say about how these maps behave? This is where immersions and submersions come in. An immersion is a smooth map $f: M \to N$ whose differential (or Jacobian) is injective at every point. The differential $dfp: Tp M \to T{f(p)} N$ is a linear map between tangent spaces. When $dfp$ is injective, it means the map is locally "stretching out" the source manifold without collapsing directions. Geometrically, immersions "locally embed" the source manifold into the target. The classic example is a circle immersed in the plane with a figure-eight shape—locally it looks like an embedding, but globally the circle crosses itself. A submersion is the dual concept: a smooth map $f: M \to N$ whose differential is surjective at every point. This means the map is "maximally surjective" at the tangent space level. Submersions often have nice properties like defining fibre bundles (we'll see this leads to covering spaces and other structures). The key difference: immersions are locally one-to-one, while submersions are locally onto. Embeddings An embedding is an immersion that is also a homeomorphism onto its image. This is the nicest possible situation: the map $f: M \to N$ is injective (one-to-one), and its inverse $f^{-1}: f(M) \to M$ is continuous. Intuitively, $f$ "nicely sits" $M$ inside $N$ without any self-intersections or pathological behavior. Not every immersion is an embedding (the figure-eight is not), but every embedding is an immersion. The subtlety is that an immersion might fail to be injective globally, even though it's locally one-to-one. Whitney's Embedding Theorem is a fundamental result: any smooth $n$-dimensional manifold can be embedded in $\mathbb{R}^{2n}$. This is remarkable because it says we never need more than twice the dimension of the manifold to find room to embed it in Euclidean space. Even more surprising, Whitney's Immersion Theorem states that any smooth $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n-1}$—we can often do even better with immersions! These theorems tell us that abstract manifolds are not so abstract after all: they all sit inside (or at least immerge into) Euclidean spaces. Isometries and Riemannian Structure When a manifold is equipped with a Riemannian metric, we can ask whether a map preserves distances. An isometry between Riemannian manifolds is a smooth map $f: (M, g) \to (N, h)$ that preserves the metric: $f^h = g$, meaning the pullback of the metric on $N$ equals the metric on $M$. Equivalently, isometries preserve all distances between points. Isometries are the "rigid motions" of Riemannian geometry—they don't stretch or compress distances at all. A related but more flexible concept is a Riemannian submersion: a smooth submersion $f: M \to N$ that is an orthogonal projection on tangent spaces. More precisely, the horizontal vectors (those orthogonal to the fibre) have their lengths preserved by the differential $df$. Riemannian submersions often arise naturally from quotient constructions, such as when we quotient a Lie group by a closed subgroup with a compatible metric. The study of isometries leads directly to the concept of Killing vector fields—vector fields $X$ that generate isometries (formally, flows of $X$ are isometries). The collection of all isometries of a Riemannian manifold forms a group (the isometry group), and Killing vector fields are precisely the infinitesimal generators of this group. Covering Spaces A covering space is a manifold $\tilde{M}$ together with a surjective smooth map $p: \tilde{M} \to M$ such that each point in $M$ has a neighborhood $U$ that is evenly covered: $p^{-1}(U)$ is a disjoint union of open sets, each homeomorphic to $U$ via $p$. Covering maps are actually local diffeomorphisms (locally invertible smooth maps), but they may fail to be invertible globally because the covering space can "wrap around" the base space multiple times. The fundamental example is the exponential map $p: \mathbb{R} \to S^1$ given by $p(x) = e^{2\pi i x}$. Here $\mathbb{R}$ covers the circle infinitely many times. Why are covering spaces important? They provide a powerful way to understand the fundamental group $\pi1(M)$: the group of homotopy classes of loops based at a point. Covering space theory gives us the lifting property: given a path or homotopy in the base space and a starting point in the covering space above the path's starting point, there is a unique lift of the path to the covering space. This relationship between coverings and the fundamental group is one of the deepest connections in topology. <extrainfo> A ramified covering generalizes this by allowing a finite set of branch points where the covering behavior degenerates. For example, the map $z \mapsto z^n$ from the complex plane to itself covers the plane $n$-to-1 everywhere except at the origin (the branch point), where all $n$ preimages collapse to a single point. Ramified coverings are useful for constructing new manifolds with specified branching behavior and appear naturally in algebraic geometry and complex analysis. </extrainfo> Scalar-Valued Functions on Manifolds Now we turn to studying functions $f: M \to \mathbb{R}$ on manifolds. These scalar-valued functions provide a rich source of geometric and topological information. Differential and Gradient Before studying special types of functions, we need to set up calculus on manifolds. The differential (or exterior derivative) of a smooth function $f: M \to \mathbb{R}$ is a covector field $df$ that measures how $f$ changes. Formally, $dfp: Tp M \to \mathbb{R}$ is given by $dfp(v) = v(f)$, the directional derivative of $f$ in direction $v$. If the manifold $(M, g)$ has a Riemannian metric, we can convert covectors to vectors via the metric. The gradient $\nabla f$ is the unique vector field satisfying: $$\langle \nabla f, X \rangle = df(X)$$ for all vector fields $X$. Geometrically, the gradient points in the direction of steepest increase of $f$, with magnitude equal to the rate of increase. The Hessian (or Hessian tensor) $\mathrm{Hess}f$ measures second-order variation of $f$. More precisely, it's the covariant derivative of the gradient: $\mathrm{Hess}f(X, Y) = (\nablaX \nabla) f(Y) = \nablaX (\nabla f)(Y)$. At a critical point where $\nabla f = 0$, the Hessian encodes all the local curvature information about $f$. Critical points occur where $df = 0$, or equivalently, where $\nabla f = 0$. A critical point is non-degenerate if the Hessian is non-singular at that point—roughly, if $f$ has a well-defined "shape" (minimum, maximum, or saddle) at that point. Morse Functions and Morse Theory A Morse function is a smooth function $f: M \to \mathbb{R}$ whose critical points are all non-degenerate. The remarkable power of Morse functions comes from Morse theory: the geometry of $f$ (specifically, its level sets) encodes the topology of $M$. For example, away from critical values, the level sets are all diffeomorphic to each other. As we pass through a critical value, the level set topology changes in a predictable way determined by the index of the critical point (the number of directions in which the Hessian is negative). More precisely, at a critical point of index $k$, the level set passes through a $k$-dimensional handle. Using this handle-attachment process, we can recover all homology information about $M$: $$\chi(M) = \sum{p \text{ critical}} (-1)^{\mathrm{index}(p)}$$ This formula connecting function critical points to topological invariants (Euler characteristic) is one of the most beautiful results in geometry. Harmonic Functions and the Laplace-Beltrami Operator The Laplace-Beltrami operator $\Delta$ is the generalization of the Euclidean Laplacian $\partial^2/\partial x1^2 + \cdots + \partial^2/\partial xn^2$ to arbitrary Riemannian manifolds. In local coordinates with respect to a Riemannian metric, it's defined as: $$\Delta f = g^{ij} \nablai \nablaj f$$ where indices are summed and $g^{ij}$ is the inverse metric. A harmonic function is a smooth function satisfying the Laplace equation: $$\Delta f = 0$$ Harmonic functions have special properties: they satisfy a maximum principle (cannot have local extrema in the interior of a domain), and they are rigid—a harmonic function on a compact manifold with no boundary must be constant. On the sphere $S^n$, the spherical harmonics are the eigenfunctions of the Laplace operator: functions satisfying $\Delta f = \lambda f$ for some eigenvalue $\lambda$. They form a complete orthogonal basis and are fundamental in mathematical physics, appearing in solutions to wave equations, heat equations, and quantum mechanics. <extrainfo> The heat equation $\partialt u = \Delta u$ governs diffusion processes on manifolds. Solutions are given as superpositions of eigenfunctions of the Laplacian. The spectrum (set of eigenvalues) of the Laplacian contains deep geometric information. The famous "hearing the shape of a drum" problem asks whether the spectrum of the Laplacian on a domain determines its geometry uniquely. The answer is "almost": spectral invariants determine much of the geometry (volume, surface area), but some non-isometric domains can have the same spectrum. The Atiyah-Singer Index Theorem, a profound result connecting analysis and topology, relates the index (dimension of kernel minus dimension of cokernel) of elliptic operators like the Laplacian to topological invariants of the manifold (characteristic classes). This theorem unifies results from analysis, topology, and algebraic geometry. </extrainfo> Other Important Function Classes Bump functions are smooth functions with compact support—they are zero outside some compact set. These are invaluable for constructing partitions of unity: collections of smooth bump functions that sum to 1 and allow us to patch together local constructions into global ones. Every manifold admits smooth partitions of unity, and this is essential for proving many important theorems. Convex functions on manifolds extend the Euclidean notion of convexity. A function is convex if it lies below its geodesic interpolations: if you take any two points and connect them by a geodesic, the function values along that geodesic should lie below the straight-line interpolation. Convex functions on manifolds appear in optimization theory and have nice properties for computing minima. <extrainfo> Morse-Bott functions generalize Morse functions by allowing critical sets to be manifolds rather than isolated points. For example, a function could have a critical circle or critical torus. The Morse-Bott condition requires that the Hessian restricted to the critical manifold is non-degenerate in the normal directions. Morse-Bott theory extends Morse theory's topological applications to this more general setting and has applications in symplectic geometry and dynamical systems. </extrainfo>