Differential geometry

The space of Taylor-equivalence classes $j_x^{k}f$ of smooth maps $M^m\to N^n$, with derivatives through order $k$ agreeing at the source point. The projection $j_x^{k}f\mapsto (x,f(x))$ has total dimension $m+n\left(\genfrac{}{}{0ex}{}{m+k}{k}\right)$ and fiber dimension $n((\genfrac{}{}{0ex}{}{m+k}{k})-1)$. First-order fibers are canonically $\mathrm{Hom}(T_{x}M,T_{y}N)$; higher-order fibers have coordinate-dependent Taylor descriptions. The truncation to order $k-1$ is an affine bundle modeled on ${\mathrm{Sym}}^k(T_x^{\ast }M)\otimes T_{y}N$.

A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

A submanifold is a manifold sitting inside another manifold through an immersion or embedding. An embedded submanifold has the subspace topology and local coordinates in which it is a coordinate plane. Smooth transverse intersections are governed by the tangent spaces of the participating submanifolds.

Morse theory relates the critical points of a Morse function to changes in the homotopy type of its sublevel sets. Crossing a nondegenerate critical value attaches a cell or handle of dimension equal to the Morse index. The theory also applies to geodesic energy through broken-geodesic approximations.

On a complete compact Riemannian manifold, nondegenerate critical points of fixed-endpoint geodesic energy yield a CW complex model of the path space, with one cell per geodesic and dimension equal to its Morse index. Finite-dimensional broken-geodesic models prove the assertion at bounded energy, and exhaustion gives the full path space. On $S^n$ with distinct nonantipodal endpoints the cell dimensions are $j(n-1)$, one in each such dimension.

A Morse-Bott critical manifold is a smooth submanifold $C$ of critical points such that the kernel of the Hessian matrix at every $x\in C$ is exactly $T_{x}C$. The Hessian is therefore nondegenerate in normal directions. Repeated based great-circle geodesics on a round sphere form such critical manifolds, parametrized by their unit initial directions.

The Morse index of a fixed-endpoint geodesic equals the number of conjugate points strictly between its endpoints, counted with multiplicity. Its nullity equals the multiplicity of the terminal conjugate point. A conjugate multiplicity is the dimension of the space of Jacobi fields vanishing at the initial point and at the point in question.

On the unit round sphere $S^n$, a geodesic of speed $L$ on $[0,1]$ has normal Jacobi equation $y^{\prime \prime }+L^{2}y=0$. Its conjugate parameters are $j\pi /L$, each with multiplicity $n-1$. If $L/\pi$ is not an integer, its index is $(n-1)\lfloor L/\pi \rfloor$. For $L=m\pi >0$, its index is $(m-1)(n-1)$ and its nullity is $n-1$.

A pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.

A Ricci-flat manifold has vanishing Ricci tensor. This is weaker than vanishing Riemann curvature tensor: the latter describes complete local flatness. A Kerr black hole supplies a Lorentzian manifold that is Ricci flat in its vacuum region but has nonzero Riemann curvature tensor.

Curvature measures the rate at which a curve or surface normal changes. A plane circle of radius $R$ has curvature $1/R$.

For an oriented plane curve written locally as the graph $(x,u(x))$, its signed curvature isThe equation $\kappa =0$ describes straight lines, while $\kappa =1/R$ describes consistently oriented arcs of circles of radius $R$.

A Morse function is a smooth real-valued function whose critical points are all nondegenerate. Near a critical point of index $k$, the Morse lemma gives coordinates in which the function is a constant minus $k$ squares plus the remaining positive squares.

The Morse lemma gives coordinates $f=f(z)-|u{|}^2+|v{|}^2$. Choose a Riemannian metric Euclidean in these coordinates. A downward gradient flow converging to $z$ has $u=0$ eventually, while $v(t)=e^{-2(t-t_0)}v(t_0)$. Thus its Riemannian distance from $z$ decays exponentially. A partition of unity makes this a global metric choice; changing the metric may change the trajectories.

The Morse index of a nondegenerate critical point is the number of negative directions of its Hessian matrix, counted with multiplicity. Equivalently, it is the maximal dimension of a subspace on which the second variation is negative definite. The same definition applies to a finite-index critical point of an infinite-dimensional energy.

The Morse lemma puts a smooth function near a nondegenerate critical point into a quadratic normal form.

A rank-$k$ distribution on a smooth manifold is a rank-$k$ vector subbundle of its tangent bundle. It is involutive when the Lie bracket of vector fields of any two local sections is again a section.

A smooth distribution is involutive if its local smooth sections are closed under the Lie bracket of vector fields. The Frobenius theorem identifies this algebraic condition with being an integrable distribution.

A distribution is integrable when every point lies on an immersed submanifold whose tangent spaces equal the distribution. Such submanifolds are its integral manifolds.

An integral manifold of a rank-$r$ smooth distribution $D$ is an immersed $r$-dimensional submanifold with $T_{p}N=D_p$ at every point. For an integrable distribution, maximal connected integral manifolds form its leaves.

The map descends to an injective immersion from $\mathbb{R}\times (\mathbb{R}/\mathbb{Z})$ into the three-dimensional torus. Injectivity follows because an integer difference in $s$ with an integer difference in $\alpha s$ must be zero. Its image is dense by an irrational rotation of the circle, and is tangent to the commuting independent fields ${\partial }_x+\alpha {\partial }_z$ and ${\partial }_y$. It is a leaf of a regular foliation, but cannot be an embedded submanifold: in an embedding chart the coordinate plane would be locally closed yet contain a dense ambient subset.

A smooth curve tangent to a regular integrable distribution lies in one maximal leaf of a regular foliation. In a Frobenius theorem chart its transverse coordinates have zero derivative, so each short segment lies in one plaque. A finite chain of such segments covers any compact parameter interval and joins plaques in the same leaf. A single embedded local plaque only gives confinement while the curve remains in its chart. A dense immersed cylinder in a three-dimensional torus shows why global embedded confinement cannot generally replace the immersed statement.

If $D={\bigcap }_{i=1}^q\ker {\alpha }^i$ has constant rank, the Frobenius theorem says that $D$ is integrable exactly when its annihilator is closed under the exterior derivative: equivalently,This is also equivalent to involutivity of $D$.

For a nowhere-zero 1-form $\alpha$, the hyperplane distribution $\ker \alpha$ is tangent to hypersurfaces exactly when $\alpha \wedge d\alpha =0$. In general relativity this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.

A nowhere-zero differential one-form $n$ is hypersurface orthogonal if locally $n=Fdt$ with $F\ne 0$. Its kernel consists of tangent vectors to the level hypersurfaces of $t$. Frobenius theorem makes this equivalent to $n\wedge dn=0$, where $d$ is the exterior derivative. With a torsion-free Levi-Civita connection, this is $n_{[\alpha }{\nabla }_{\beta }{n}_{\gamma ]}=0$. For a timelike unit normal, projecting the derivative on both indices gives zero antisymmetric part, which explains why hypersurface orthogonality implies symmetric extrinsic curvature.

For a non-null Killing vector field $V$ of a Levi-Civita connection, hypersurface orthogonality implies that $\eta =V^{♭}/F$, with $F=g(V,V)$, is a closed differential form. The Killing equation and $V^{♭}\wedge d{V}^{♭}=0$ give $2F{\nabla }_{\mu }{V}_{\nu }=V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F$, whence $d\eta =0$. The Poincare lemma gives $\eta =d\varphi$ locally. For a timelike $V$, $\varphi$ supplies local static time coordinates. This does not assert global exactness.

For an involutive distribution of constant rank $r$, induct on $r$. Straighten a nonzero section by the flow-box theorem, and choose a frame ${\partial }_1,Y_2,\dots ,Y_r$ with no ${\partial }_1$ component in the remaining fields. Involutivity implies ${\partial }_{1}Y=AY$ for their column $Y$. The invertible solution of ${\partial }_{1}B=-BA$, initialized by $B=I$ on $x_1=0$, makes $Z=BY$ independent of $x_1$. Its restriction to the transverse slice is involutive of rank $r-1$, so induction provides coordinates spanning it. Extending these coordinates independently of $x_1$ gives $D=\mathrm{span}({\partial }_1,\dots ,{\partial }_r)$. Conversely, fields tangent to an integral manifold have tangent brackets because they preserve the ideal of smooth functions vanishing on it.

A plane distribution on a three-manifold is a rank-two distribution. Locally it is the kernel of a nowhere-zero differential 1-form $\alpha$.

The Frobenius theorem gives the local criterion

A contact structure on a three-manifold is a plane distribution locally of the form $\ker \alpha$ with $\alpha \wedge d\alpha$ nowhere zero. It is maximally nonintegrable; the standard example on ${\mathbb{R}}^3$ is $\ker (dz-xdy)$.

A smooth interval family of contact forms on a compact manifold without boundary is carried back to its initial contact distribution by a smooth isotopy. The pullback of each form is a positive smooth multiple of the initial form. Solve for the horizontal generator of contact stability and integrate the resulting scalar equation for the contact conformal factor along an isotopy.

If a time-dependent vector field satisfies ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, its flow maps satisfy ${\rho }_t^{\ast }{\alpha }_t=u_t{\alpha }_0$, where $u_t(x)=\exp ({\int }_0^t{h}_s({\rho }_s(x))ds)$. Thus the conformal factor is smooth and positive even when $h_t$ changes sign.

For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.

A contact form $\alpha$ defines the hyperplane distribution $\xi =\ker \alpha$. Multiplying $\alpha$ by a nowhere-zero function leaves this distribution unchanged. The contact distribution symplectic form distinguishes it from an integrable hyperplane distribution.

For a contact form on a $(2r+1)$-manifold, $d\alpha {|}_{\ker \alpha }$ is nondegenerate: the contact condition says its top exterior power is nowhere zero on the hyperplanes. This fiberwise symplectic form identifies contact-plane vectors with covectors on that plane and determines the horizontal generator of contact stability.

A contact form on a $(2n+1)$-manifold is a one-form satisfying $\alpha \wedge (d\alpha {)}^n\ne 0$ everywhere.

The Reeb vector field of a contact form is uniquely characterized by

For ${\alpha }_s=\frac{1}{2}(r_1^{2}d{\theta }_1+s{r}_2^{2}d{\theta }_2)$ on $S^3$, the Reeb field is $2{\partial }_{{\theta }_1}+2s^{-1}{\partial }_{{\theta }_2}$. If $s$ is irrational, its only closed orbits are the two coordinate circles.

A smooth map is a submersion when its differential is surjective at every point.

A surjective submersion between Riemannian manifolds is Riemannian when its derivative is a linear isometry from the orthogonal complement of its kernel onto the target tangent space. Its vertical space is the kernel and its horizontal space is the orthogonal complement.

For a Riemannian submersion with totally geodesic submanifolds as fibers, the positive Laplace-Beltrami operator commutes with pullback of functions. Horizontal terms in the Riemannian Hessian are pulled back from the base, and vertical terms vanish. Minimal fibers suffice because only the trace of the vertical second fundamental form enters.

A basic function for a Riemannian submersion is a function pulled back from the base. It is constant along each fiber. Its Riemannian gradient is the horizontal lift of a vector field through a submersion of the base gradient. Constancy on connected fiber components alone need not imply global descent when fibers are disconnected.

If $F:X^n\to Y^m$ is a submersion at $p$, there are local coordinates in whichEvery fiber of a submersion is consequently an embedded submanifold of codimension $m$.

After choosing a Riemannian metric on the source of a submersion $F:X\to Y$, the orthogonal complements $H_p=(\ker D_{p}F{)}^{\perp }$ map isomorphically to $T_{F(p)}Y$. Therefore every vector field $w$ on $Y$ has the smooth lift

The fibers of a proper submersion over a connected base are mutually diffeomorphic. Lift a compactly supported vector field moving one nearby base point to another; properness makes the lifted field compactly supported, and its time-one flow identifies the fibers.

An immersion is a smooth map whose differential is injective at every point. A parametrized surface in ${\mathbb{R}}^3$ is immersed when its two coordinate tangent vectors are linearly independent.

An isometric immersion is an immersion $f:M\to N$ between Riemannian manifolds whose derivative preserves the inner products of tangent vectors; equivalently, $f^{\ast }{g}_N=g_M$. Its derivative is injective and the target may have larger dimension. A local isometry additionally requires a local diffeomorphism, so its tangent maps are linear isomorphisms. For example, a sphere with its induced Riemannian metric is isometrically immersed in Euclidean space, although its positive intrinsic sectional curvature differs from that of the ambient space.

A smooth embedding is an injective immersion that is a homeomorphism onto its image with the subspace topology.

A tubular neighborhood of a smooth embedded submanifold is a neighborhood identified with a neighborhood of the zero section in its normal bundle. Radial deformation of punctured fibers gives the sphere bundle, and excision identifies the associated pair with a normal disk-bundle pair.

An immersed submanifold of $M$ is a manifold $N$ with an injective immersion $\iota :N\to M$; when $N$ is treated as a subset, its manifold topology can be finer than its subspace topology.

For irrational $\alpha$, the mapis an injective immersion with dense image. It is not a smooth embedding, since points with arbitrarily large parameter return arbitrarily close to its initial point in the subspace topology.