Smooth manifold

A smooth $k$-manifold is a Hausdorff second-countable space locally homeomorphic to ${\mathbb{R}}^k$, equipped with smoothly compatible coordinate charts.

A smooth embedded surface in ${\mathbb{R}}^3$ is a two-dimensional smooth manifold locally represented by an injective regular parametrized surface $\sigma :U\subset {\mathbb{R}}^2\to {\mathbb{R}}^3$, with ${\sigma }_u\times {\sigma }_v\ne 0$. The first fundamental form measures lengths and angles, and a choice of unit normal determines the second fundamental form, which measures bending. A surface with boundary additionally uses local half-plane charts along its boundary.

Every finitely presented group is the fundamental group of a closed connected smooth oriented four-manifold. Build a compact four-dimensional handle decomposition with one zero-handle, a one-handle for each generator of a group and a two-handle for each relator. Turning the handle decomposition upside down shows that its boundary surjects on the fundamental group. Its double of a manifold has the same fundamental group by the Seifert-van Kampen theorem.

Products of manifold charts form a smooth atlas on the topological product of two smooth manifolds. The transition maps are products of the two smooth transition maps, so the dimension is the sum of the dimensions. The smooth structure is independent of the chosen compatible atlases.

A band sum joins two embedded spheres or links by a narrow tube along an arc, removing a small disk from each and joining their boundaries. In a handle slide, the construction uses a parallel framed copy of the second attaching sphere and transports its framing of an embedded sphere along the band.

A compact manifold is a smooth manifold whose underlying topological space is compact. Every smooth vector field on a compact manifold without boundary has a flow defined for all real time. On a manifold with boundary, tangency to the boundary or an appropriate inward-flow condition must also be imposed.

A closed manifold is a compact manifold with boundary whose boundary is empty. Here “closed” includes compactness.

A compact two-dimensional manifold without boundary. A connected oriented surface is classified by its genus; for genus $g$ its Euler characteristic is $2-2g$.

For connected oriented closed $d$-dimensional manifolds with $d\ge 2$, their connected sum removes an open $d$-dimensional ball from each and identifies the resulting boundary spheres by an orientation-reversing diffeomorphism. The remaining orientations fit together. Collapsing the separating sphere gives a pinch map to the wedge sum of the two closed manifolds, and projection to either summand has degree of a continuous mapping one. In intermediate positive degrees the cohomology is the direct sum of the summand groups. Products of classes from different summands vanish; products landing in top degree use the single common orientation class. These statements follow from excision, the Mayer–Vietoris sequence and the degree-one projections.

Remove small disks around a nondegenerate maximum of a smooth function on one closed surface and a nondegenerate minimum on another. Rescale their values so the boundary values agree with the two ends of a neck. Their regular level collars allow a strictly monotone function on the neck, with nonzero derivative, matching smoothly to both sides. The resulting connected sum has all the old critical points except the deleted maximum and minimum. This permits one degenerate torus saddle to be retained while adding ordinary handles for higher genus.

For a connected sum of identical oriented manifolds of dimension at least three, one can permute the attachment balls in a punctured sphere and match the attached copies by their chosen parametrizations. This gives orientation-preserving diffeomorphisms realizing the permutations on the middle cohomology summands. For sums of complex projective planes, these permutations preserve the diagonal positive intersection form.

// Destination: geometry-and-topology.bigb

Tube the projective lines in the two summands of ${\mathbb{CP}}^2\#{\mathbb{CP}}^2$ through the connecting neck. Their connected sum is an embedded two-sphere in the class $h_1+h_2$, with self-intersection two. Its oriented normal disk bundle has Euler number two, so its boundary is the unit tangent bundle of $S^2$, namely $SO(3)\cong {\mathbb{RP}}^3$. The boundary separates the interior of the tubular neighborhood from its nonempty exterior.

// Destination: geometry-and-topology.bigb

For closed connected oriented $d$-manifolds with $d\ge 2$,Deleting an open ball changes the Euler characteristic by $-1+\chi (S^{d-1})$, and gluing subtracts $\chi (S^{d-1})$. Since $\chi (S^{d-1})+\chi (S^d)=2$, the formula follows. Thus a connected sum of oriented manifolds subtracts two from the sum of Euler characteristics in even dimension and subtracts zero in odd dimension.

For the standard complex orientations, the two summand classes have zero product and equal squares, the oriented degree-four class. The quadratic relations imply all cubic monomials vanish. The rational Betti numbers are $1,2,1$ in degrees zero, two and four.

For odd $r\ge 1$ and $W_g={\#}_{i=1}^g(S^r\times S^r)$, the integral cohomology has one copy of $\mathbb{Z}$ in degrees zero and $2r$, ${\mathbb{Z}}^{2g}$ in degree $r$, and zero elsewhere. Choose the top orientation class $\omega$ and degree-$r$ classes ${\alpha }_i,{\beta }_i$ from the two factors of summand $i$. Their nonzero positive-degree basis products areAll ${\alpha }_i{\alpha }_j$ and ${\beta }_i{\beta }_j$ vanish, and $\omega$ annihilates positive-degree classes. The Künneth theorem computes each sphere product; the Mayer–Vietoris sequence computes the connected sum groups, and its degree-one pinch maps determine the products. Thus its middle-degree Poincare duality pairing is a symplectic vector space after changing coefficients to a field of characteristic different from two. The case $r=1$ is the usual cohomology ring of a closed oriented surface.

A smooth manifold with boundary has charts in the half-space $\{x_n\ge 0\}\subseteq {\mathbb{R}}^n$, with smoothly compatible transition maps. The boundary is the locus modeled on $x_n=0$.

A collar neighborhood identifies a neighborhood of the boundary with $\partial W\times [0,\epsilon )$, with the boundary at parameter zero. It allows a manifold's boundary to be pushed slightly inward by a homotopy.

A parallelizable manifold is a smooth $n$-manifold whose tangent bundle is trivial, equivalently one admitting $n$ smooth vector fields that form a basis of every tangent space. Every Lie group is parallelizable because a basis at the identity extends to a global frame by left translation.

Among positive-dimensional spheres, precisely $S^1$, $S^3$, and $S^7$ are parallelizable. The circle and three-sphere inherit Lie-group frames from $U(1)$ and $SU(2)$; the seven-sphere has a frame constructed from octonion multiplication.

For $q=(q_0,q_1,q_2,q_3)\in S^3$, the vectors $(-q_1,q_0,q_3,-q_2)$, $(-q_2,-q_3,q_0,q_1)$ and $(-q_3,q_2,-q_1,q_0)$ are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating $-i$ times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.

The real $m$ by $n$ matrices of matrix rank $m\le n$ form an open smooth manifold $X_{m,n}\subseteq {\mathbb{R}}^{mn}$. Indeed, it is the union of the open sets on which a chosen $m$ by $m$ minor has nonzero determinant, so its dimension is $mn$.

An isotopy is a continuous one-parameter family of embeddings or homeomorphisms beginning at a given map.

Two embeddings $f_0,f_1:N\hookrightarrow M$ are ambient isotopic when a continuously varying family of ambient homeomorphisms $F_t$ starting at the identity satisfies $F_1{f}_0=f_1$. This is stronger than an arbitrary homotopy of maps. Smooth versions use smooth families of diffeomorphisms; unparametrized versions identify embedded images, and oriented versions preserve their orientations.

A smooth isotopy is a smooth family of smooth embeddings or diffeomorphisms parametrized by an interval and beginning at the given map.

The boundary connected sum of two manifolds with boundary removes a boundary half-ball from each and glues the resulting hemispherical faces. Its boundary is the ordinary connected sum of the original boundaries.

This boundary gluing operation is related to the connected sum, but applies along boundary disks rather than removing interior balls.

An embedded submanifold $M\subseteq N$ is a subset whose inclusion is an embedding. Around every point there are coordinates on $N$ in which $M$ is a coordinate plane.

An embedded submanifold is a submanifold for which the inclusion gives the subspace topology, unlike a merely immersed submanifold.

If an embedded codimension-$k$ submanifold $X\subset M$ is $f^{-1}(y)$ for a regular value $y$ of a smooth map $f:M\to {\mathbb{R}}^k$, then $df$ identifies its normal bundle with the trivial bundle $X\times {\mathbb{R}}^k$. A submanifold with nontrivial normal bundle therefore cannot be such a regular level set. The core circle of the Möbius band is a codimension-one example.

If $N^n\subset M^m$ is embedded and $p\in N$, there are smooth coordinates $(x^1,\dots ,x^m)$ around $p$ in whichThe first $n$ coordinates restrict to a chart on $N$.

An injectively immersed submanifold $N\subset M$ is embedded if every smooth function on $N$ extends to a smooth function on $M$. Otherwise a sequence from a different local sheet can converge in $M$ to a point $p$ while remaining outside a small embedded neighborhood of $p$ in $N$; a bump function supported on that neighborhood cannot have a continuous ambient extension.

The vanishing ideal is $I(N)=\{f\in C^{\infty }(M):f{|}_N=0\}$. A vector field $X$ is tangent to $N$ exactly when $X(I(N))\subseteq I(N)$.

If $X$ and $Y$ preserve $I(N)$, then their commutator preserves it becausefor every $f\in I(N)$. Thus vector fields tangent to an embedded submanifold are closed under the Lie bracket of vector fields, and the restricted bracket depends only on their restrictions to the submanifold.

If embedded submanifolds $A,B\subseteq M$ meet transversely, then $A\cap B$ is an embedded submanifold of dimension $\dim A+\dim B-\dim M$.

The hypothesis is transversality: tangent spaces to the two submanifolds span the ambient tangent space at each intersection point.

Submanifolds $A,B\subseteq M$ intersect cleanly when $A\cap B$ is an embedded submanifold andat every $x\in A\cap B$. A transverse intersection is the special case in which $T_{x}A+T_{x}B=T_{x}M$.

Submanifolds $A,B\subseteq M$ intersect transversely when $T_{x}A+T_{x}B=T_{x}M$ at every intersection point. The transverse intersection theorem then determines the dimension and smoothness of $A\cap B$.

For complementary-dimensional oriented submanifolds of an oriented manifold meeting transversely in finitely many points, the smooth intersection number is the sum of their local orientation signs. Geometric disjointness implies number zero, but number zero alone need not imply geometric disjointness. Without coherent orientations, use the count modulo $2$.

A hypersurface in an $n$-dimensional smooth manifold is a smooth submanifold of dimension $n-1$. Locally, a regular level set $\{\varphi =0\}$ with $\nabla \varphi \ne 0$ is a hypersurface whose normal vector is $\nabla \varphi$.

A non-null hypersurface projection annihilates the unit normal vector and fixes tangent vectors to the hypersurface. It obeys $P^2=P$. Apply it to every tensor index to remove normal components, with the metric used to lower or raise the index pair for covariant slots. For a timelike unit normal in mostly-plus signature it is the spatial projection tensor. A null normal cannot be normalized this way, and a null hypersurface requires additional choices for a corresponding projection.

Local defining functions $f_{\alpha }$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_{\alpha }/f_{\beta }$ on overlaps. These ratios define a real line bundle with a global transverse section whose local representatives are the $f_{\alpha }$ and whose zero set is $Y$. The bundle is naturally isomorphic to the normal bundle of $Y$.

The extrinsic curvature of a hypersurface is the tangential bilinear form obtained by differentiating its unit normal. It measures how the hypersurface bends in the ambient manifold, with an overall sign depending on convention.

A hypersurface is totally geodesic when every ambient geodesic initially tangent to it remains in it. Equivalently, its extrinsic curvature vanishes identically.

An abstract smooth surface is a two-dimensional smooth manifold. Thus it is a Hausdorff second-countable space with an atlas of plane-valued coordinate charts whose transition maps are smooth.

A finite group acting freely by diffeomorphisms on a smooth manifold has a smooth quotient. Choose each coordinate neighbourhood disjoint from all its nontrivial translates and transport its chart through the quotient map. The resulting transition maps are restrictions of the original transition maps composed with group elements.

A map between smooth manifolds is smooth when its expression in every pair of coordinate charts is a smooth function.

A map between smooth manifolds is smooth when its coordinate expressions in compatible charts have derivatives of every order. The condition is invariant under smooth chart changes and expresses smoothness in charts.

A continuous map from a smooth manifold to ${\mathbb{R}}^d$ can be approximated by a smooth map with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local convolution and then combining with a smooth partition of unity gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line homotopy to be supported in that domain. This permits the Sard theorem to be used after approximation; the original continuous map can be space-filling and need not omit a point.

A smooth map $f:M\to N$ is transverse to an embedded submanifold $S\subset N$ whenThis is a sum of vector subspaces, not necessarily a direct sum. Equivalently the composition of $d{f}_p$ with the quotient to $T_{f(p)}N/T_{f(p)}S$ is surjective. In a slice chart for an embedded submanifold of codimension $k$, the $k$ normal coordinate functions composed with $f$ have surjective derivative on their common zero set. If the preimage is empty the condition holds vacuously.

Smooth maps of finitely many circles into a manifold of dimension at least three can be perturbed within their homotopy classes to disjoint embeddings. The transversality dimension count for two curve branches is $1+1-m<0$, so their images avoid one another after perturbation; one also removes self-intersections and obtains immersions by the jet version of transversality. This allows disjoint representative loops for relators in high-dimensional surgery constructions of prescribed fundamental groups.

If $f:M\to N$ satisfies transversality of a map to a submanifold $S$, its nonempty preimage is an embedded submanifold of the displayed codimension. Choose a slice chart for an embedded submanifold around $f(p)$ in which $S$ is the zero set of the first $k$ coordinates. Their composition with $f$ has surjective derivative on its zero set, so the regular level set theorem gives a local embedded slice of dimension $\dim M-k$. These slices give the subspace topology and a compatible smooth atlas on the preimage. Its tangent space is

A pushforward of a curve composes a parametrized curve in the source with a smooth map between manifolds. Its tangent is the image of the original tangent under the differential of a smooth map. A constant image curve has zero tangent, so a general smooth map need not preserve regularity of curves.

A pullback of a smooth function composes a function on the target with a smooth map between manifolds, producing a function on the source. It needs no inverse map. Acting on the pulled-back function defines the differential of a smooth map intrinsically as a map of derivations.

For a smooth map between manifolds $f:M\to N$, the differential $d{f}_x:T_{x}M\to T_{f(x)}N$ is the linear map obtained by differentiating in charts. Intrinsically it sends the tangent vector of a curve $\gamma$ through $x$ to the tangent vector of $f\circ \gamma$. The chain rule makes it independent of charts and gives $d(g\circ f{)}_x=d{g}_{f(x)}\circ d{f}_x$.

For a real-valued smooth function on a smooth manifold, its differential of a smooth map is a differential one-form: $d{f}_p$ maps a tangent vector to the directional derivative of the function. In local coordinates, $df={\sum }_i({\partial }_{i}f)d{x}^i$. The exterior derivative extends this operation from functions to higher-degree differential forms.

A smooth map between smooth manifolds with zero differential is locally constant: each target coordinate function has zero derivatives on a sufficiently small source coordinate ball. Integration along straight segments proves constancy there. Its fibers are open and closed, so a connected source makes the map constant globally.

The pushforward of a contravariant tensor applies the differential of a smooth map to each vector factor of a type $(r,0)$ tensor. It is defined pointwise at a specified source point. A field generally gives a field along the map, not necessarily a unique tensor field on the image when several points have the same image. A diffeomorphism allows the usual transport of arbitrary mixed tensors through its inverse as well.

The pullback of a covariant tensor composes each of its vector arguments with the differential of a smooth map. It takes a type $(0,s)$ tensor on the target to one on the source. Antisymmetry is not required; the pullback of a differential form is the alternating special case. If the target vectors are projected onto the image tangent space first, evaluating the pullback is unchanged.

The pullback of a covector is the dual linear map to the differential of a smooth map. In coordinates its components are $({\varphi }^{\ast }\eta {)}_i=({\partial }_i{y}^{\alpha }){\eta }_{\alpha }$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.

For a vector field $X$ on $M$, its pushforward by a smooth map between manifolds $f:M\to N$ is the vector field along a map defined by $(f_{\ast }X{)}_p=d{f}_p(X_p)$. Acting on a function $h$ on $N$, it satisfies $(f_{\ast }X{)}_p(h)=X_p(h\circ f)$. For a diffeomorphism this defines a vector field on $N$ by evaluation at $p=f^{-1}(q)$. For a general map there need not be a single vector at each image point: $f(x)=x^2$, $X={\partial }_x$ gives $2x{\partial }_y$, with opposite values over $y>0$.

The chain rule shows that $\alpha {\varphi }_t{\alpha }^{-1}$ is the local flow of the pushforward of a vector field ${\alpha }_{\ast }X$. If this equals $X$, uniqueness of integral curves of a vector field gives commutation. Conversely, differentiate the commuting identity at $t=0$ to obtain $d{\alpha }_p{X}_p=X_{\alpha (p)}$. All identities hold on their common domains; the vector field need not be complete.

A vector field $X$ is projectable through a smooth map between manifolds $f:M\to N$ if there is a smooth vector field $Y$ on $N$ with $d{f}_p{X}_p=Y_{f(p)}$ for every $p$. Agreement on fibres is necessary. For a surjective submersion it is also sufficient: smooth local sections of the submersion express $Y$ locally as $df(X)$ and prove its smoothness. An arbitrary non-surjective map may instead give a field only along its image, with extension to $N$ requiring additional choices.

A local flow of a smooth vector field $V$ is a smooth map $\Phi :D\to M$, defined on an open neighborhood of $\{0\}\times M$, such that ${\Phi }^0(x)=x$, ${\partial }_t{\Phi }^t(x)=V({\Phi }^t(x))$, and ${\Phi }^{s+t}={\Phi }^s\circ {\Phi }^t$ whenever both sides are defined.

A smooth vector field is complete if its maximal integral curves of a vector field exist for every real time. Its local flow is then a global one-parameter group of diffeomorphisms; uniqueness of ordinary differential equations proves the group law and the inverse-time identity. On a compact manifold without boundary, every smooth vector field is complete: compactness gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb{R}$, $x^2{\partial }_x$ is not complete because its solution $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

On ${\mathbb{R}}^2$, the vector fields $X=y{\partial }_x$ and $Y=(x^2/2){\partial }_y$ are complete, since their local flows are $(x+ty,y)$ and $(x,y+t{x}^2/2)$. Their Lie bracket of vector fields is $-(x^2/2){\partial }_x+xy{\partial }_y$. Its integral curve of a vector field starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an integral curve of a vector fieldthat cannot continue through $t=2\sqrt{3}$. Thus neither operation preserves complete vector fields on a general noncompact smooth manifold. This contrasts with a compact manifold without boundary, where every smooth vector field is complete.

Near a point where a smooth vector field is nonzero, there are coordinates in which it is $\partial /\partial x_1$. Flow a transverse coordinate slice by the local flow; the inverse function theorem makes the resulting map a coordinate chart. This is the first step of the coordinate proof of the Frobenius theorem.

Every open cover of a smooth paracompact manifold admits a smooth partition of unity subordinate to it: nonnegative smooth functions with locally finite supports in the cover whose sum is one.

For each $T>0$ there is a nonnegative test function $\Psi$ with the displayed identity. Choose a nonnegative bump function $\eta$ supported in $[-T,T]$ and positive on $[-T/2,T/2]$. The sum $S(x)={\sum }_k\eta (x+kT)$ is a positive, smooth, $T$ periodic function. Setting $\Psi =\eta /S$ preserves compact support and smoothness, and periodicity of $S$ proves ${\sum }_k\Psi (x+kT)=S(x)/S(x)=1$. The translates form a locally finite partition of unity on the real line.

A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.

This atlas criterion expresses orientability for a differentiable manifold.

An orientation of a smooth manifold is a continuous choice of orientation of a vector space in every tangent space, equivalently an orientation of a vector bundle on its tangent bundle. It can be specified by an atlas with positive Jacobian determinants on all overlaps or by a nowhere-vanishing top-degree differential form. On a connected orientable manifold there are two choices, interchanged by reversing every tangent-space orientation.

// Target: differential-geometry.bigb

An oriented atlas is a smooth atlas whose smooth transition maps have positive Jacobian determinants on every overlap. The coordinate bases then give compatible choices of orientation of a vector space on all tangent spaces. A nowhere-vanishing top-degree differential form gives such an atlas: restrict each chart until its coefficient has constant sign, reverse one coordinate if necessary to make that sign positive, and compare the positive coefficients on overlaps. Conversely, a partition of unity glues positive local coordinate top forms to a global volume form.

The boundary of an oriented manifold with boundary is oriented by the outward-normal-first convention: an ordered basis $(v_2,\dots ,v_n)$ of $T_p\partial X$ is positive when $(v_1,v_2,\dots ,v_n)$ is positive in $T_{p}X$ for an outward-pointing normal $v_1$.

For a smooth $n$-manifold, the following are equivalent: an atlas with positive transition Jacobians, a smooth choice of orientation of every tangent space, and a nowhere-vanishing smooth $n$-form. Positive charts orient their coordinate frames; oriented frames select the positive local coordinate volume forms, which a partition of unity glues; and a nonzero top form declares a frame positive when its value on that frame is positive.

If $M\times N$ is orientable, fix a point of $N$ and an ordered basis of its tangent space. Contracting a product orientation with that basis gives a smooth orientation of $M$. Interchanging the factors gives an orientation of $N$.

An orientation-reversing diffeomorphism reverses the chosen orientation. In oriented local coordinates its derivative has negative determinant.