A circle can be drawn in the plane, but the plane is not part of what makes the circle a one-dimensional smooth object. Near each point of the circle, a small arc behaves like an interval, so calculus ought to make sense there. The problem is that no single interval coordinate describes the whole circle without either repeating a point or cutting the circle open. Smooth manifolds solve this by replacing global coordinates with compatible local coordinates. The local pieces look like open subsets of Euclidean space, and the compatibility condition says that moving from one coordinate description to another is governed by ordinary smooth changes of variables. In this way, multivariable calculus becomes available on spaces that are globally curved, wrapped, or assembled from several coordinate patches. [example: Why One Coordinate Does Not Cover $S^1$] Let $S^1=\{(x,y)\in\mathbb R^2:x^2+y^2=1\}$, and define $\gamma:(0,2\pi)\to S^1$ by $\gamma(t)=(\cos t,\sin t)$. Since $\cos$ and $\sin$ are smooth real functions, both coordinate functions of $\gamma$ are smooth. Also $\gamma(t)\in S^1$ because \begin{align*} \cos^2 t+\sin^2 t=1. \end{align*} The point $(1,0)$ is not in the image, since $\cos t=1$ and $\sin t=0$ force $t=0$ or $t=2\pi$ modulo $2\pi$, neither of which lies in $(0,2\pi)$. Conversely, every point of $S^1\setminus\{(1,0)\}$ has a unique polar angle $t\in(0,2\pi)$, so $\gamma$ parametrizes exactly the punctured circle. This parametrization cannot be made into one coordinate chart on all of $S^1$ by merely adding an endpoint. If $0$ and $2\pi$ were both allowed, then \begin{align*} \gamma(0)=(\cos 0,\sin 0)=(1,0) \end{align*} and \begin{align*} \gamma(2\pi)=(\cos 2\pi,\sin 2\pi)=(1,0), \end{align*} so the map would not be one-to-one. If one instead keeps an open interval as the coordinate domain, then one endpoint of the interval is missing, and that missing endpoint is precisely the point where the circle closes. The local-coordinate remedy is to use more than one chart. Let $N=(0,1)$ and $S=(0,-1)$, and define stereographic coordinates \begin{align*} \varphi_N(x,y)=\frac{x}{1-y} \end{align*} on $S^1\setminus\{N\}$ and \begin{align*} \varphi_S(x,y)=\frac{x}{1+y} \end{align*} on $S^1\setminus\{S\}$. For $t\in\mathbb R$, the inverse of $\varphi_N$ is \begin{align*} \varphi_N^{-1}(t)=\left(\frac{2t}{1+t^2},\frac{t^2-1}{1+t^2}\right), \end{align*} because \begin{align*} \left(\frac{2t}{1+t^2}\right)^2+\left(\frac{t^2-1}{1+t^2}\right)^2=\frac{4t^2+(t^2-1)^2}{(1+t^2)^2}=\frac{t^4+2t^2+1}{(1+t^2)^2}=1. \end{align*} On the overlap $S^1\setminus\{N,S\}$, the coordinate $t=\varphi_N(x,y)$ is nonzero, and the transition map is \begin{align*} (\varphi_S\circ\varphi_N^{-1})(t)=\frac{\frac{2t}{1+t^2}}{1+\frac{t^2-1}{1+t^2}}=\frac{\frac{2t}{1+t^2}}{\frac{2t^2}{1+t^2}}=\frac{1}{t}. \end{align*} The reverse transition is the same formula with the variables renamed, so both transition maps are smooth on $\mathbb R\setminus\{0\}$. Thus $S^1$ is not naturally described by one global interval coordinate; it is described by compatible local coordinates whose changes of coordinate are ordinary smooth functions. [/example] This local viewpoint changes what differentiability means. A real-valued function on the circle is smooth when its expression in each allowed coordinate is smooth. A map between curved spaces is smooth when all its coordinate representatives are smooth Euclidean maps. The theory therefore needs a precise way to say which coordinate systems are allowed and how they agree. ## Definition A smooth manifold is the object that keeps exactly the coordinate information needed for calculus and discards the accidental coordinates used to describe it. The definition packages two ingredients: a space that is locally Euclidean, and a maximal rule saying which local coordinates are smoothly compatible with one another. [definition: Smooth Manifold] A smooth $n$-manifold is a pair $(M,\mathcal A)$ where $M$ is a topological $n$-manifold and $\mathcal A$ is a maximal smooth atlas on $M$. [/definition] When the atlas is understood, one writes $M$ rather than $(M,\mathcal A)$. The boundaryless smooth manifold is the parent notion here; a smooth manifold with boundary changes the local Euclidean model and appears later as a related specialization. The rest of this section unpacks the supporting notions that make the compact definition usable. ## Atlases and Coordinate Compatibility ### Local Euclidean Spaces Before smoothness can be defined, the underlying space must at least have Euclidean neighbourhoods. The Hausdorff condition prevents distinct points from being topologically inseparable, while second countability keeps the space within the usual scope of local-to-global arguments. These hypotheses isolate the spaces on which coordinate calculus has a reasonable chance of behaving like ordinary calculus. [definition: Topological $n$-Manifold] A topological $n$-manifold is a second-countable Hausdorff [topological space](/page/Topological%20Space) $M$ such that for every point $p\in M$ there exist an open neighbourhood $U$ of $p$ in $M$ and a homeomorphism $\varphi:U\to\varphi(U)$ onto an open subset of $\mathbb R^n$. [/definition] A topological manifold gives the stage, but calculus needs named local measuring devices. We must be able to refer to a chosen neighbourhood and its coordinate map as a single object, because all later compatibility and smoothness tests are phrased in terms of such local measurements. [definition: Coordinate Chart] Let $M$ be a topological $n$-manifold. A coordinate chart on $M$ is a pair $(U,\varphi)$ where $U$ is open in $M$ and $\varphi:U\to\varphi(U)$ is a homeomorphism onto an open subset of $\mathbb R^n$. [/definition] If $(U,\varphi)$ is a chart and $\varphi(p)=(x_1(p),\ldots,x_n(p))$, the functions $x_i:U\to\mathbb R$ are local coordinate functions. A chart by itself is too small to describe most manifolds, and many charts may overlap. The next definition answers the essential question: when do two coordinate systems describe the same differentiable structure? [definition: Smoothly Compatible Charts] Let $M$ be a topological $n$-manifold. Two coordinate charts $(U,\varphi)$ and $(V,\psi)$ on $M$ are smoothly compatible if either $U\cap V=\varnothing$, or the transition maps $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ and $\varphi\circ\psi^{-1}:\psi(U\cap V)\to\varphi(U\cap V)$ are smooth maps between open subsets of $\mathbb R^n$. [/definition] Smooth compatibility is the mechanism that makes coordinate calculations independent of the chosen chart. Still, a manifold usually needs a cover by many charts rather than an isolated compatible pair. We therefore gather compatible charts into a single coordinate system covering the whole space. [definition: Smooth Atlas] Let $M$ be a topological $n$-manifold. A smooth atlas on $M$ is a collection $\mathcal A=\{(U_i,\varphi_i):i\in I\}$ of coordinate charts such that the sets $U_i$ cover $M$ and every two charts in $\mathcal A$ are smoothly compatible. [/definition] An atlas can often be described using only a few charts, but there are many additional charts compatible with those choices. To make the smooth structure independent of this presentation, we need a canonical enlargement that contains every chart allowed by the same compatibility rule. [definition: Maximal Smooth Atlas] Let $M$ be a topological $n$-manifold. A maximal smooth atlas on $M$ is a smooth atlas $\mathcal A$ such that every coordinate chart on $M$ that is smoothly compatible with every chart in $\mathcal A$ already belongs to $\mathcal A$. [/definition] The maximal atlas records all coordinate systems admitted by the smooth structure. In practice, however, we almost never list it directly; we generate it from a smaller atlas whose transition maps are manageable. ### Generating the Structure The definition uses maximal atlases, but examples are rarely presented by listing all compatible charts. What we need is a theorem saying that a manageable compatible atlas already determines the full smooth structure. This is the practical bridge between examples and the formal definition. [quotetheorem:3901] The theorem means that checking transition maps for a convenient covering family is enough. Once those checks are done, all compatible charts are included automatically. This is how standard examples are constructed in practice. [example: The Smooth Structure on $S^1$] Let $S^1\subseteq\mathbb R^2$ have the [subspace topology](/page/Subspace%20Topology), and write $N=(0,1)$ and $S=(0,-1)$. Put $U_N=S^1\setminus\{N\}$ and $U_S=S^1\setminus\{S\}$. Since points are closed in $\mathbb R^2$, the sets $U_N$ and $U_S$ are open in $S^1$, and they cover $S^1$. Define \begin{align*} \varphi_N(x,y)=\frac{x}{1-y} \end{align*} on $U_N$ and \begin{align*} \varphi_S(x,y)=\frac{x}{1+y} \end{align*} on $U_S$. For $t\in\mathbb R$, set \begin{align*} \varphi_N^{-1}(t)=\left(\frac{2t}{1+t^2},\frac{t^2-1}{1+t^2}\right). \end{align*} This point lies on $S^1$ because \begin{align*} \left(\frac{2t}{1+t^2}\right)^2+\left(\frac{t^2-1}{1+t^2}\right)^2=\frac{4t^2+t^4-2t^2+1}{(1+t^2)^2}=\frac{t^4+2t^2+1}{(1+t^2)^2}=1. \end{align*} It is not $N$, since its second coordinate equals $1$ would imply $t^2-1=1+t^2$, hence $-1=1$. Also \begin{align*} \varphi_N(\varphi_N^{-1}(t))=\frac{\frac{2t}{1+t^2}}{1-\frac{t^2-1}{1+t^2}}=\frac{\frac{2t}{1+t^2}}{\frac{2}{1+t^2}}=t. \end{align*} Conversely, if $(x,y)\in U_N$, then $x^2+y^2=1$ and $y\ne1$. With $t=x/(1-y)$, we have \begin{align*} \frac{2t}{1+t^2}=\frac{\frac{2x}{1-y}}{1+\frac{x^2}{(1-y)^2}}=\frac{2x(1-y)}{(1-y)^2+x^2}. \end{align*} Using $x^2=1-y^2=(1-y)(1+y)$, the denominator becomes \begin{align*} (1-y)^2+x^2=(1-y)^2+(1-y)(1+y)=2(1-y). \end{align*} Therefore $2t/(1+t^2)=x$. Similarly, \begin{align*} \frac{t^2-1}{1+t^2}=\frac{\frac{x^2}{(1-y)^2}-1}{1+\frac{x^2}{(1-y)^2}}=\frac{x^2-(1-y)^2}{x^2+(1-y)^2}. \end{align*} The numerator is \begin{align*} x^2-(1-y)^2=(1-y)(1+y)-(1-y)^2=2y(1-y), \end{align*} and the denominator is $2(1-y)$, so $(t^2-1)/(1+t^2)=y$. Thus $\varphi_N$ is a chart with inverse as displayed. The same calculation for the south chart gives \begin{align*} \varphi_S^{-1}(s)=\left(\frac{2s}{1+s^2},\frac{1-s^2}{1+s^2}\right). \end{align*} On $U_N\cap U_S$, the coordinate $t=\varphi_N(x,y)$ is nonzero, because $\varphi_N^{-1}(0)=S$, which is not in the overlap. Hence the transition map has domain $\mathbb R\setminus\{0\}$, and \begin{align*} (\varphi_S\circ\varphi_N^{-1})(t)=\frac{\frac{2t}{1+t^2}}{1+\frac{t^2-1}{1+t^2}}=\frac{\frac{2t}{1+t^2}}{\frac{2t^2}{1+t^2}}=\frac{1}{t}. \end{align*} The reverse transition is also $s\mapsto1/s$ on $\mathbb R\setminus\{0\}$. Since $t\mapsto1/t$ is smooth on $\mathbb R\setminus\{0\}$, the two stereographic charts are smoothly compatible. They therefore form a smooth atlas on $S^1$, and by the *Atlas [Extension Theorem](/theorems/59)* this atlas generates the usual maximal smooth structure on the circle. [/example] The circle demonstrates the general pattern: a manifold is often recognized by finding enough local coordinates and checking their overlaps. The same pattern will appear for spheres, matrix groups, and regular level sets. ## Smooth Maps and Diffeomorphisms After defining smooth spaces, the next problem is defining smooth maps between them. A map $F:M\to N$ cannot be differentiated directly in the ambient sense unless both manifolds already sit inside Euclidean spaces. The coordinate strategy is to express $F$ in charts and demand ordinary smoothness there. [definition: Smooth Map Between Smooth Manifolds] Let $M$ be a smooth $m$-manifold and let $N$ be a smooth $n$-manifold. A map $F:M\to N$ is smooth if $F$ is continuous and for every $p\in M$, every chart $(V,\psi)$ on $N$ with $F(p)\in V$, there exists a chart $(U,\varphi)$ on $M$ with $p\in U$ and $F(U)$ contained in $V$ such that the coordinate representation $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V)$ is smooth. [/definition] This definition is intentionally phrased in all charts, so it makes coordinate independence part of the statement. In calculations, however, checking every possible pair of charts would be unusable. The practical question is whether it is enough to test the coordinate expression on charts from selected smooth atlases, because those are the charts actually available in examples and constructions. [quotetheorem:9872] This result is why manifold arguments feel local. The global assertion that $F$ is smooth can be checked by covering the source and target with convenient coordinate patches. The next notion identifies when two smooth manifolds are the same object from the perspective of smooth calculus. [definition: Diffeomorphism] Let $M$ and $N$ be smooth manifolds. A diffeomorphism from $M$ to $N$ is a bijective smooth map $F:M\to N$ whose inverse $F^{-1}:N\to M$ is smooth. [/definition] The inverse condition is essential. A smooth bijection may fail to preserve differentiability when read backward, so it cannot serve as an isomorphism of smooth manifolds. The following example is the standard warning. [example: Smooth Bijection Without Smooth Inverse] Consider $F:\mathbb R\to\mathbb R$ defined by $F(t)=t^3$. Since $F$ is a polynomial function, it is smooth in the usual Euclidean sense. It is injective because if $F(a)=F(b)$, then \begin{align*} a^3-b^3=0 \end{align*} and \begin{align*} a^3-b^3=(a-b)(a^2+ab+b^2). \end{align*} The second factor satisfies \begin{align*} a^2+ab+b^2=\left(a+\frac b2\right)^2+\frac{3b^2}{4}\ge0, \end{align*} and it is zero only when $a=b=0$. Hence $a=b$, so $F$ is injective. It is surjective because for every $s\in\mathbb R$, the real number $t=s^{1/3}$ satisfies \begin{align*} F(t)=F(s^{1/3})=(s^{1/3})^3=s. \end{align*} Thus $F$ is a smooth bijection with inverse $F^{-1}(s)=s^{1/3}$. The inverse is not differentiable at $0$. Its difference quotient at $0$ is \begin{align*} \frac{F^{-1}(h)-F^{-1}(0)}{h}=\frac{h^{1/3}-0}{h}=\frac{1}{h^{2/3}} \end{align*} for $h\ne0$. As $h\to0$, the quantity $1/h^{2/3}$ is positive and unbounded, so the derivative of $F^{-1}$ at $0$ does not exist as a finite real number. Therefore a diffeomorphism must require smoothness of both the map and its inverse. [/example] Diffeomorphisms let us ignore accidental presentations. The same smooth manifold may be embedded, parametrized, or described by equations in many ways, but its smooth structure is preserved exactly by diffeomorphisms. ## Constructing Smooth Manifolds ### Euclidean Models and Level Sets The simplest smooth manifolds are already present in multivariable calculus. Every open subset of Euclidean space is a smooth manifold, and these examples are the local models for all others. This grounding is useful because every manifold definition must reduce to the usual one in coordinates. [example: Open Subsets of Euclidean Space] Let $U$ be open in $\mathbb R^n$. The single chart $(U,\operatorname{id}_U)$ covers $U$, and its only transition map is \begin{align*} \operatorname{id}_U\circ \operatorname{id}_U^{-1}=\operatorname{id}_U\circ \operatorname{id}_U=\operatorname{id}_U. \end{align*} The identity map on an open subset of $\mathbb R^n$ is smooth, so this one-chart atlas is a smooth atlas and therefore generates a maximal smooth atlas on $U$. With this smooth structure, the coordinate expression of a map $f:U\to\mathbb R^m$ in the identity charts is \begin{align*} \operatorname{id}_{\mathbb R^m}\circ f\circ \operatorname{id}_U^{-1}=\operatorname{id}_{\mathbb R^m}\circ f\circ \operatorname{id}_U=f. \end{align*} Thus if $f$ is smooth in the usual multivariable sense, then its identity-coordinate representative is smooth, so $f$ is smooth as a map of manifolds. Conversely, suppose $f$ is smooth as a map of manifolds. For each $p\in U$, choose a chart $(W,\varphi)$ around $p$ such that $f\circ\varphi^{-1}$ is smooth. Since $(W,\varphi)$ belongs to the maximal atlas generated by the identity chart, it is smoothly compatible with $(U,\operatorname{id}_U)$, so \begin{align*} \varphi=\varphi\circ\operatorname{id}_U^{-1} \end{align*} is smooth on $W$, and $\varphi^{-1}$ is smooth on $\varphi(W)$. On $W$ we have \begin{align*} f=(f\circ\varphi^{-1})\circ\varphi. \end{align*} The right-hand side is a composition of ordinary smooth maps between open subsets of Euclidean spaces, so $f$ is ordinarily smooth near $p$. Since $p$ was arbitrary, $f$ is smooth on $U$ in the usual multivariable sense. Thus the manifold notion of smoothness on open Euclidean subsets recovers exactly the standard one. [/example] Open subsets are the local models, but many geometric spaces appear as subsets of a larger Euclidean space. To recognize such a subset as a manifold, we need a definition that says it can be straightened locally into a coordinate plane. This motivates the embedded submanifold condition. [definition: Embedded Submanifold] Let $M$ be a subset of $\mathbb R^N$. The subset $M$ is an embedded $n$-submanifold of $\mathbb R^N$ if for every $p\in M$ there exist an [open set](/page/Open%20Set) $W$ in $\mathbb R^N$ with $p\in W$, an open set $O$ in $\mathbb R^N$, and a diffeomorphism $\Phi:W\to O$ such that $\Phi(W\cap M)=O\cap(\mathbb R^n\times\{0\}^{N-n})$. [/definition] This definition captures the absence of cusps, crossings, and rank loss. It remains to know when equations produce such subsets. The regular level set theorem supplies the main construction criterion. Here the equations are packaged as a smooth map $F:M\to\mathbb R^k$. At a point $p$, the derivative or differential $dF_p:T_pM\to\mathbb R^k$ is the best linear approximation to $F$ on tangent vectors. The condition that a value be regular means that $dF_p$ has rank $k$ at every point of the level set $F^{-1}(c)$. Its kernel $\ker dF_p$ is the space of tangent directions that stay inside the constraint surface to first order, and codimension $k$ means that the constraint removes $k$ independent local directions. The key question is therefore whether this first-order rank condition is strong enough to control the actual shape of the zero set. The regular level set theorem answers this by turning independent equations into local coordinates, so that the solution set becomes a genuine embedded submanifold with the expected dimension. [quotetheorem:6837] The rank condition says that the equations remove exactly $k$ independent directions. When the derivative loses rank, the equation may describe a singular space rather than a manifold. [example: The Sphere as a Regular Level Set] Let $F:\mathbb R^{n+1}\to\mathbb R$ be defined by \begin{align*} F(x_1,\ldots,x_{n+1})=|x|^2=x_1^2+\cdots+x_{n+1}^2. \end{align*} Then \begin{align*} F^{-1}(1)=\{x\in\mathbb R^{n+1}:x_1^2+\cdots+x_{n+1}^2=1\}=S^n. \end{align*} The function $F$ is smooth because it is a polynomial in the coordinate functions. For $p=(p_1,\ldots,p_{n+1})$ and $h=(h_1,\ldots,h_{n+1})$, expand \begin{align*} F(p+h)=\sum_{i=1}^{n+1}(p_i+h_i)^2=\sum_{i=1}^{n+1}p_i^2+2\sum_{i=1}^{n+1}p_i h_i+\sum_{i=1}^{n+1}h_i^2. \end{align*} Since $F(p)=\sum_{i=1}^{n+1}p_i^2$, the first-order part in $h$ is \begin{align*} DF_p(h)=2\sum_{i=1}^{n+1}p_i h_i. \end{align*} If $p\in S^n$, then $\sum_{i=1}^{n+1}p_i^2=1$, so $p\ne0$. To see that $DF_p:\mathbb R^{n+1}\to\mathbb R$ is surjective, let $r\in\mathbb R$ and choose $h=(r/2)p$. Then \begin{align*} DF_p(h)=2\sum_{i=1}^{n+1}p_i\left(\frac r2 p_i\right)=r\sum_{i=1}^{n+1}p_i^2=r. \end{align*} Thus every point of $F^{-1}(1)$ is regular. By the *Regular Level Set Theorem*, $S^n=F^{-1}(1)$ is an embedded smooth submanifold of $\mathbb R^{n+1}$ of dimension $(n+1)-1=n$. [/example] The sphere is smooth because its defining equation has nonzero first-order variation along the level set. The next example shows what the theorem is protecting us from. [example: A Cone Point That Is Not a Smooth Manifold] Let \begin{align*} C=\{(x,y,z)\in\mathbb R^3:x^2+y^2=z^2\} \end{align*} and let $F:\mathbb R^3\to\mathbb R$ be \begin{align*} F(x,y,z)=x^2+y^2-z^2. \end{align*} Then $C=F^{-1}(0)$, and \begin{align*} DF_{(x,y,z)}(a,b,c)=2xa+2yb-2zc. \end{align*} At the origin this becomes \begin{align*} DF_{(0,0,0)}(a,b,c)=0a+0b-0c=0, \end{align*} so $DF_{(0,0,0)}:\mathbb R^3\to\mathbb R$ is the zero map and is not surjective. If $p=(x,y,z)\in C$ and $p\ne(0,0,0)$, then not all of $x,y,z$ are zero, so the linear functional $DF_p$ is nonzero. A nonzero linear functional $\mathbb R^3\to\mathbb R$ is surjective: if, for example, $x\ne0$, then for any $r\in\mathbb R$, \begin{align*} DF_p\left(\frac{r}{2x},0,0\right)=2x\frac{r}{2x}+2y\cdot0-2z\cdot0=r, \end{align*} and the cases $y\ne0$ and $z\ne0$ are identical using the second or third coordinate. Thus the regular level set argument applies away from the origin, but fails exactly at the cone point. The failure is also visible topologically. In $C\setminus\{(0,0,0)\}$ the equation $x^2+y^2=z^2$ implies $z\ne0$, because $z=0$ would give $x^2+y^2=0$, hence $x=y=0$. Therefore \begin{align*} C\setminus\{(0,0,0)\}=(C\cap\{z>0\})\cup(C\cap\{z<0\}), \end{align*} a union of two disjoint nonempty relatively open pieces. Every neighbourhood of the origin in $C$ contains points from both pieces, for instance $(\varepsilon,0,\varepsilon)$ and $(\varepsilon,0,-\varepsilon)$ for sufficiently small $\varepsilon>0$, so deleting the origin disconnects that neighbourhood into the two nappes. By contrast, deleting a point from a small open disk in $\mathbb R^2$ leaves a connected punctured disk. Indeed, after translating the deleted point to $0$, any two nonzero points in a disk can be joined by polygonal line segments chosen to avoid $0$. Connectedness is preserved by homeomorphisms, so no neighbourhood of the cone point can be homeomorphic to an open subset of $\mathbb R^2$. The origin is therefore not a manifold point, even though the cone is smooth away from it. [/example] ### Matrix Groups Equations also build manifolds inside spaces of matrices. This matters because many geometric transformation groups are simultaneously algebraic objects and smooth manifolds. The determinant constraint is the simplest useful model. [example: The Special Linear Group] Identify the [vector space](/page/Vector%20Space) of real $n\times n$ matrices with $\mathbb R^{n^2}$. The determinant is a polynomial in the matrix entries, hence continuous, so \begin{align*} GL(n,\mathbb R)=\det^{-1}(\mathbb R\setminus\{0\}) \end{align*} is open in $\mathbb R^{n^2}$. The special linear group is \begin{align*} SL(n,\mathbb R)=\{A\in GL(n,\mathbb R):\det A=1\}=\det^{-1}(1). \end{align*} Let $A\in SL(n,\mathbb R)$ and let $H$ be an arbitrary real $n\times n$ matrix. Put $B=A^{-1}H$. For real $s$ near $0$, \begin{align*} \det(A+sH)=\det(A(I+sA^{-1}H))=\det(A)\det(I+sB). \end{align*} Using the permutation formula for the determinant, \begin{align*} \det(I+sB)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^n(\delta_{i,\sigma(i)}+sB_{i,\sigma(i)}). \end{align*} The constant term comes from the identity permutation and equals $1$. The coefficient of $s$ also comes only from the identity permutation, because every non-identity permutation moves at least two indices and therefore contributes either no constant term or terms of degree at least $2$ in $s$. Thus \begin{align*} \det(I+sB)=1+s\sum_{i=1}^nB_{ii}+s^2R(s) \end{align*} for some polynomial $R(s)$. Since $\sum_{i=1}^nB_{ii}=\operatorname{tr}(B)$, we get \begin{align*} \frac{\det(A+sH)-\det(A)}{s}=\det(A)\left(\operatorname{tr}(A^{-1}H)+sR(s)\right). \end{align*} Letting $s\to0$ gives \begin{align*} D(\det)_A(H)=\det(A)\operatorname{tr}(A^{-1}H). \end{align*} Because $A\in SL(n,\mathbb R)$, $\det(A)=1$, so \begin{align*} D(\det)_A(H)=\operatorname{tr}(A^{-1}H). \end{align*} This [linear map](/page/Linear%20Map) $D(\det)_A:\mathbb R^{n^2}\to\mathbb R$ is surjective. Indeed, for any $r\in\mathbb R$, choose $H=(r/n)A$. Then \begin{align*} D(\det)_A\left(\frac rn A\right)=\operatorname{tr}\left(A^{-1}\frac rn A\right)=\operatorname{tr}\left(\frac rn I_n\right)=\frac rn\operatorname{tr}(I_n)=\frac rn\cdot n=r. \end{align*} Hence $1$ is a regular value of the determinant. By the *Regular Level Set Theorem*, $SL(n,\mathbb R)=\det^{-1}(1)$ is an embedded smooth submanifold of $\mathbb R^{n^2}$ of dimension $n^2-1$. At the identity matrix, \begin{align*} D(\det)_I(H)=\operatorname{tr}(I^{-1}H)=\operatorname{tr}(H), \end{align*} so the tangent space is \begin{align*} T_I SL(n,\mathbb R)=\ker D(\det)_I=\{H\in M_n(\mathbb R):\operatorname{tr}(H)=0\}. \end{align*} Thus the first-order directions in $SL(n,\mathbb R)$ at the identity are exactly the trace-zero matrices. [/example] This example foreshadows Lie groups: smooth manifolds with group operations compatible with the smooth structure. The smooth manifold supplies the calculus; the group law supplies symmetry. ## Tangent Spaces and Local Linearization ### Derivations at a Point Calculus on manifolds needs a coordinate-free meaning of tangent direction. A curve through a point gives one intuition, but an intrinsic vector should act on smooth functions by taking directional derivatives. This leads to the derivation definition. [definition: Tangent Vector as a Derivation] Let $M$ be a smooth manifold and let $p\in M$. A tangent vector at $p$ is an $\mathbb R$-linear map $v:C^\infty(M)\to\mathbb R$ such that for all $f,g\in C^\infty(M)$, $v(fg)=f(p)v(g)+g(p)v(f)$. The tangent space $T_pM$ is the vector space of all tangent vectors at $p$. [/definition] This definition makes a tangent vector into a first-order differential operator at a point. To compare tangent vectors under a smooth map, we need a derivative that transports such operators from the source to the target. [definition: Differential of a Smooth Map] Let $F:M\to N$ be a smooth map between smooth manifolds, and let $p\in M$. The differential of $F$ at $p$ is the linear map $dF_p:T_pM\to T_{F(p)}N$ defined by $(dF_pv)(f)=v(f\circ F)$ for every $v\in T_pM$ and every $f\in C^\infty(N)$. [/definition] In coordinates, this is represented by the [Jacobian matrix](/page/Jacobian%20Matrix) of the coordinate expression of $F$. The definition avoids choosing coordinates, but it agrees with the usual derivative when charts are chosen. ### Bundles and Fields Tangent spaces vary from point to point, and many geometric objects require choosing a tangent vector at every point smoothly. To make that precise, the separate vector spaces $T_pM$ must be assembled into a single geometric object. [definition: Tangent Bundle] Let $M$ be a smooth manifold. The tangent bundle of $M$ is the disjoint union $TM=\bigsqcup_{p\in M}T_pM$ together with the projection $\pi:TM\to M$ sending each $v\in T_pM$ to $p$. [/definition] The tangent bundle should itself be a smooth manifold, since its points are positions together with velocities. Local coordinates on $M$ provide position coordinates and tangent-vector components, but on overlaps those components transform by Jacobian matrices. The issue is whether these transformation rules are smooth enough to make the disjoint union $TM$ into a manifold in a way compatible with the projection to $M$. [quotetheorem:3908] Once the tangent bundle is a smooth manifold, we can ask for smooth maps into it that choose exactly one tangent vector over each base point. This is the right global version of a first-order direction field, and it motivates the definition of a smooth vector field. [definition: Smooth Vector Field] Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth map $X:M\to TM$ such that $\pi\circ X=\operatorname{id}_M$. [/definition] In a chart $(U,\varphi)$ with coordinates $(x_1,\ldots,x_n)$, a vector field has the local expression $X=\sum_{i=1}^n X_i\partial_{x_i}$, where the coefficient functions $X_i:U\to\mathbb R$ are smooth. This is the form used in coordinate computations. ## Rank, Fibres, and Global Tools ### Immersions and Submersions The differential of a smooth map contains its first-order geometry. If it is injective, the map preserves tangent directions; if it is surjective, the map has enough independent output directions to behave locally like a projection. These two rank conditions organize much of the local theory. [definition: Immersion] Let $M$ and $N$ be smooth manifolds, and let $F:M\to N$ be smooth. The map $F$ is an immersion if for every $p\in M$, the linear map $dF_p:T_pM\to T_{F(p)}N$ is injective. [/definition] Immersions prevent first-order collapse but may still have global self-intersections. The complementary local problem asks when a map has full rank onto its target, so that its fibres behave like regular level sets. This motivates the submersion condition. [definition: Submersion] Let $M$ and $N$ be smooth manifolds, and let $F:M\to N$ be smooth. The map $F$ is a submersion if for every $p\in M$, the linear map $dF_p:T_pM\to T_{F(p)}N$ is surjective. [/definition] A submersion should have level sets that are smooth manifolds, with tangent spaces given by kernels of differentials. The obstacle is that a fibre $F^{-1}(q)$ is defined by an equation inside an arbitrary manifold, not inside a fixed Euclidean space. Surjectivity of $dF_p$ is the condition that lets charts straighten the map near each point of the fibre, turning the fibre locally into a coordinate plane. The natural question is therefore whether this local straightening can be assembled into an intrinsic manifold structure on each regular fibre. The regular level set theorem gives the needed bridge from the rank condition on $F$ to the smooth structure and tangent-space description of $F^{-1}(q)$. [quotetheorem:9873] This theorem explains why kernels of differentials have geometric meaning. They are not only linear algebra data; they are the tangent spaces to fibres. ### Partitions of Unity Smooth manifolds are defined locally, but many constructions require gluing local data. Without a controlled gluing device, coordinate-level definitions would remain local fragments. Partitions of unity are the main tool that turns local smooth constructions into global smooth objects. [definition: Support of a Smooth Function] Let $M$ be a smooth manifold and let $f:M\to\mathbb R$ be a smooth function. The support of $f$ is $\operatorname{supp}f=\overline{\{p\in M:f(p)\ne0\}}$. [/definition] Support records where a smooth function is active. To glue local constructions, we need families of smooth functions supported inside prescribed coordinate neighbourhoods and arranged so that only finitely many terms affect any one point. This motivates partitions of unity. [definition: Smooth Partition of Unity] Let $M$ be a smooth manifold and let $\mathcal U=\{U_i:i\in I\}$ be an [open cover](/page/Open%20Cover) of $M$. A smooth [partition of unity](/page/Partition%20of%20Unity) subordinate to $\mathcal U$ is a family $\{\rho_j:j\in J\}$ of smooth functions $\rho_j:M\to\mathbb R$, together with a choice of index $i(j)\in I$ for each $j\in J$, such that $0\le\rho_j\le1$, $\operatorname{supp}\rho_j$ is contained in $U_{i(j)}$, the family of supports is locally finite, and $\sum_{j\in J}\rho_j(p)=1$ for every $p\in M$. [/definition] The indexing set for the partition need not be the same as the indexing set for the original cover. This flexibility matters: the usual construction first refines the cover, then builds bump functions whose closed supports sit inside chosen cover elements. The next theorem is the global existence result that makes this gluing method available. [quotetheorem:3917] A central application is the construction of Riemannian metrics. Local coordinates provide local Euclidean inner products, and partitions of unity blend them into a global smoothly varying [inner product](/page/Inner%20Product). Before measuring lengths and angles on a manifold, each tangent space needs an inner product that varies smoothly from point to point. The notation $T^*M$ denotes the cotangent bundle, whose fiber at $p$ is the dual vector space $T_p^*M$ of linear functions on $T_pM$. A smooth section of a bundle is a smooth choice of one element in each fiber. Thus a section of $\operatorname{Sym}^2(T^*M)$ is a smoothly varying symmetric covariant 2-tensor: at each point it takes two tangent vectors and returns a real number, symmetrically and bilinearly. Positive definite means that this [bilinear form](/page/Bilinear%20Form) is an inner product on every tangent space. [definition: Riemannian Metric] Let $M$ be a smooth manifold. A Riemannian metric on $M$ is a smooth section $g:M\to\operatorname{Sym}^2(T^*M)$ of the bundle of symmetric covariant $2$-tensors such that, for each $p\in M$, the tensor $g_p:T_pM\times T_pM\to\mathbb R$ is a positive definite inner product. [/definition] A Riemannian metric is extra structure, not part of the definition of a smooth manifold. Locally there is no difficulty: a chart imports the Euclidean dot product. The real question is whether these local inner products can be blended on overlaps without losing smoothness or positive definiteness, and partitions of unity provide exactly the required averaging device. [quotetheorem:6096] This theorem is the doorway from smooth topology to Riemannian geometry. It also shows how the local-to-global machinery of smooth manifolds produces major geometric structures. ## Boundaries, Orientations, and Forms The definition of smooth manifold uses open subsets of $\mathbb R^n$ as local models. Spaces such as intervals, disks, and compact regions have edge points, where the correct local model is a half-space rather than an open Euclidean set. This motivates the standard boundary variant. [definition: Smooth Manifold with Boundary] A smooth $n$-manifold with boundary is a second-countable Hausdorff topological space $M$ equipped with a maximal atlas of charts $\varphi:U\to\varphi(U)$, where $\mathbb H^n=\{x\in\mathbb R^n:x_n\ge0\}$ and each $\varphi(U)$ is relatively open in $\mathbb H^n$. The transition maps between overlapping charts are smooth in the half-space sense: near each point of their domain, they extend to smooth maps on open subsets of $\mathbb R^n$. [/definition] Boundary points are represented by coordinate vectors with last coordinate $0$. This is a specialization of the local-model idea rather than the same definition as a boundaryless smooth manifold. Integration on manifolds requires another global choice: a consistent sign convention for coordinates. Without it, local volume forms may disagree by sign on overlaps. This is the role of orientation. [definition: Orientation] Let $M$ be a smooth $n$-manifold with maximal smooth atlas $\mathcal A$. An orientation on $M$ is a choice of subatlas $\mathcal A^+$ of $\mathcal A$ that covers $M$, is maximal among subatlases whose transition maps have positive Jacobian determinant, and is compatible with the fixed smooth structure $\mathcal A$. [/definition] Orientation prepares the manifold for integration. The objects being integrated are differential forms, which are designed to transform correctly under coordinate changes and therefore fit naturally with the atlas formalism. The cotangent bundle also provides the language for integration on manifolds. For a vector space $V$, $\Lambda^k V^*$ denotes the space of alternating $k$-linear functions on $V$, meaning multilinear functions that change sign when two inputs are swapped and vanish when two inputs agree. Applying this to each tangent space gives the bundle $\Lambda^kT^*M$. The notation $\Gamma(E)$ means the smooth sections of a bundle $E$, so $\Omega^k(M)=\Gamma(\Lambda^kT^*M)$ is the space of smooth choices of an alternating covariant $k$-tensor at every point of $M$. To make this coordinate-free object precise, we need a definition that records two conditions at once: the value at each point must live in the correct exterior power over that point, and those values must vary smoothly across the manifold. This is exactly what a smooth section of $\Lambda^kT^*M$ encodes. [definition: Differential $k$-Form] Let $M$ be a smooth manifold, and let $\pi:\Lambda^kT^*M\to M$ be the bundle of alternating covariant $k$-tensors. A differential $k$-form on $M$ is a smooth map $\omega:M\to\Lambda^kT^*M$ such that $\pi\circ\omega=\operatorname{id}_M$. The space of smooth differential $k$-forms is denoted $\Omega^k(M)=\Gamma(\Lambda^kT^*M)$. [/definition] Differential forms connect local calculus to global topology. To see why they matter, we need the theorem that relates differentiating a form on a manifold to integrating the form over the boundary. This is the central global calculus statement on oriented manifolds. [quotetheorem:3900] [Stokes' theorem](/theorems/1530) shows why smooth manifolds are a natural setting for global calculus. It contains the [fundamental theorem of calculus](/theorems/632), [Green's theorem](/theorems/3612), and the [divergence theorem](/theorems/2754) as manifestations of one geometric statement. ## Beyond and Connected Topics Smooth manifolds are the base layer for much of modern geometry. Adding a Riemannian metric leads to [Riemannian geometry](/page/Cambridge%20III%20Riemannian%20Geometry), where lengths, geodesics, curvature, and comparison theorems become central. [Differential forms and de Rham cohomology](/page/Differential%20Forms%20and%20de%20Rham%20Cohomology) continue the story in [Differential Forms II: Manifolds and Cohomology](/page/Differential%20Forms%20II%3A%20Manifolds%20and%20Cohomology). There the focus shifts from local coordinates to global invariants detected by closed and exact forms. Embedded and immersed submanifolds connect this page to [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Tangent spaces, vector fields, rank theorems, and Lie brackets become the working language for curvature and flows. Algebraic geometry offers a contrasting viewpoint. In [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry), spaces are often defined by polynomial equations, and singularities are central objects rather than exceptional failures. Lie groups are smooth manifolds equipped with compatible group operations. Their tangent spaces at the identity become Lie algebras, turning nonlinear group structure into linear infinitesimal structure. Manifolds with boundary and corners are indispensable for integration domains, moduli spaces, and variational problems. They preserve the local-coordinate philosophy while allowing controlled edge behaviour. ## References Androma, [Differential Forms II: Manifolds and Cohomology](/page/Differential%20Forms%20II%3A%20Manifolds%20and%20Cohomology). Androma, [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). John M. Lee, *Introduction to Smooth Manifolds* (2013). Victor Guillemin and Alan Pollack, *Differential Topology* (1974). Frank W. Warner, *Foundations of Differentiable Manifolds and Lie Groups* (1983).