Riemannian geometry

Riemannian geometry studies smooth manifolds equipped with a Riemannian metric: an inner product at each tangent space, varying smoothly with position. This local measuring rule determines angles, arc length, Riemannian distance and geodesics. An inhabitant of a surface can discover its geometry by measuring within it, without knowing how it sits in a larger Euclidean space.

A circular cylinder is a particularly useful example. Its surface bends in space, yet unrolling it preserves the induced metric: a small patch has the geometry of a flat sheet. By contrast, a round sphere has positive Gaussian curvature. The distinction between intrinsic flatness of a circular cylinder and extrinsic curvature separates the geometry an inhabitant measures from the way an outside observer sees the surface bend.

Geodesics supply the analogue of straight lines. On a round sphere they follow great circles; on a circular cylinder they become straight lines when lifted to an unrolled sheet. Their arc length lets us compare candidate routes, as in great-circle distance and Riemannian distance on a circular cylinder. The same measuring rule also defines differential operators such as the Laplace-Beltrami operator, connecting geometry with diffusion and spectral geometry.

Bernhard Riemann's 1854 inaugural lecture proposed studying spaces of arbitrary dimension through their infinitesimal measuring rules, including a positive quadratic expression for the squared line element. This moved the subject beyond a particular surrounding space: the measuring rule itself became part of the geometry. ^[ref].

A minimal submanifold has vanishing mean curvature vector, equivalently zero first variation of volume for compactly supported normal variations. A minimal surface is its two-dimensional case. A calibrated submanifold satisfies the stronger property of minimizing volume in its relative homology class.

// Target: general-relativity.bigb

A Riemannian metric satisfying the displayed equation for a vector field $X$ and constant $\lambda$. A gradient Ricci soliton has $X=\nabla f$; it is steady when $\lambda =0$.

For $\mathrm{Ric}+{\nabla }^{2}f=0$, the contracted Bianchi identity and the commutation identity $\Delta \nabla f=\nabla \Delta f+\mathrm{Ric}(\nabla f)$ imply $\nabla R=2\mathrm{Ric}(\nabla f)$. Meanwhile $\nabla |\nabla f{|}^2=-2\mathrm{Ric}(\nabla f)$, so their sum is constant on every connected component.

The symmetric covariant two-tensor defined using the Levi-Civita connection by ${\mathrm{Hess}}_{g}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Its negative metric trace is the positive Laplace-Beltrami operator applied to $f$. Along a geodesic it gives the second derivative of $f$.

Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

For the positive Laplace-Beltrami operator $P$ on a closed manifold, the wave trace is the distribution ${\sum }_j{e}^{it\sqrt{{\lambda }_j}}$, counting multiplicities. The series is interpreted after testing against smooth compactly supported functions, rather than as an ordinarily convergent sum. Its singularities relate the spectrum to the length spectrum of closed geodesics, through the Duistermaat-Guillemin trace formula.

For the positive Laplace-Beltrami operator on a closed manifold, the singular support of the wave trace lies in zero together with signed lengths of closed geodesics. Under the clean fixed-set hypotheses, its singularity at a periodic-geodesic length has an expansion determined by the geodesic flow and its linearized return map. In particular, when the relevant leading coefficient does not vanish through cancellation, that length is determined by the spectrum. The inclusion of singular supports alone does not justify an unconditional recovery of every geodesic length.

For the positive scalar Laplace-Beltrami operator on a closed $d$-dimensional Riemannian manifold, the number of eigenvalues at most $\lambda$, counted with multiplicity, has the displayed leading asymptotic. The exponent determines the dimension and the coefficient determines the total volume of the Riemannian metric. There is an analogous result with fixed Dirichlet boundary conditions or Neumann boundary conditions on a compact smooth domain.

For a nonnegative elliptic operator $L$ on a closed manifold, sum its positive eigenvalues with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim \ker L$ and $K(t)=\mathrm{Tr}(e^{-tL})$ is the heat trace, thenSubtracting finitely many terms of the short-time heat kernel expansion gives a meromorphic continuation. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.

For a Laplace-type operator on a closed manifold, let $a_{t^0}$ be the constant coefficient of its full heat trace expansion. In the Mellin transform defining the spectral zeta function, the reduced trace has constant coefficient $a_{t^0}-\dim \ker L$. Its contribution is $(a_{t^0}-\dim \ker L)/s$, and $1/\Gamma (s)=s+O(s^2)$ proves the identity. For the scalar positive Laplace-Beltrami operator on a closed surface, $a_{t^0}=\int RdV/(24\pi )$ and $\dim \ker L$ is the number of connected components. In particular, a connected sphere has value $-2/3$ and a connected flat torus has value $-1$.

For the positive Laplace-Beltrami operator on a closed compact Riemannian manifold, the heat trace is $Z_M(t)={\sum }_j{e}^{-t{\lambda }_j}={\int }_M{K}_M(t,x,x)dV(x)$. It records every eigenvalue with multiplicity. In a finite free isometric quotient it equals $|U{|}^{-1}{\sum }_{u\in U}{\int }_N{K}_N(t,x,ux)dV(x)$. The orbital integrals form a class function of the ambient finite isometry group, so Gassmann equivalence gives equal quotient heat traces and hence Sunada theorem.

For a compact smooth Riemannian surface with smooth boundary and a pure Dirichlet boundary condition or pure Neumann boundary condition, the constant coefficient in the heat trace isThe equality is the Gauss-Bonnet theorem. Thus such isospectral manifolds have the same Euler characteristic. A simply connected compact connected bordered surface is a disk with $\chi =1$, whereas a nonorientable surface with $c\ge 1$ crosscaps and $b\ge 1$ boundary components has $\chi =2-c-b\le 0$. These cannot be isospectral in this setting. Corners and mixed boundary conditions contribute additional heat coefficients, so the smooth pure-boundary-condition hypothesis is essential.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

For a smooth ambient function near the unit sphere, the chain rule along the great circles ${\gamma }_i(t)=p\cos t+e_i\sin t$ gives $(F\circ {\gamma }_i{)}^{\prime \prime }(0)=D^{2}F(e_i,e_i)-DF(p)$. Sum over an orthonormal basis of the tangent space and supplement it by the radial vector $p$. The geodesic trace formula for the Laplace-Beltrami operator then gives the displayed formula, evaluated at radius one. In particular a degree-$\ell$ harmonic polynomial restricts to a spherical harmonic with positive eigenvalue $\ell (\ell +d-1)$.

Two Riemannian manifolds are isospectral for a specified differential operator if their eigenvalues, counted with multiplicities and with the same boundary conditions, agree. Riemannian isometries preserve the Laplace-Beltrami operator spectrum, but the converse can fail. The transplantation theorem and the Sunada theorem give systematic constructions.

For a fixed closed orientable Riemannian surface, the smooth metrics with a specified scalar Laplace-Beltrami operator spectrum form a compact set modulo diffeomorphisms, in the smooth topology. This is a compactness assertion, not uniqueness or finiteness of the number of Riemannian isometry classes. It restricts geometric degeneration within an isospectral set and uses more information than the leading heat invariants alone. The theorem was established in the 1988 compactness paper.

A pair of closed orientable flat three-manifolds with equal scalar Laplace-Beltrami operator spectra but different first Betti numbers, respectively one and zero. Consequently they are not homeomorphic. They cannot be isospectral for the Hodge Laplacian on one-forms, whose zero-eigenvalue multiplicity is the first Betti number. This shows why the spectrum on functions must not be silently replaced by the spectra on all differential forms.

For a fixed closed hyperbolic surface, there are only finitely many isometry classes with its Laplacian spectrum. The Selberg trace formula gives a common discrete length spectrum and common positive hyperbolic systole. Mumford's compactness theorem confines this family to a compact subset of moduli space. If distinct members accumulated, choose nearby markings at an accumulation point and use finite length coordinates for hyperbolic surfaces. Each of these finitely many lengths converges while remaining in one discrete spectrum, so it is eventually constant. The length coordinates then force eventual equality, a contradiction. This finiteness theorem is weaker than the Wolpert generic spectral rigidity theorem.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

For a finite subgroup $U$ of a group $T$ of Riemannian isometries of a closed manifold $N$, assume $U$ acts freely and define $I_t(g)={\int }_N{H}_N(t,x,gx)dV(x)$. Invariance of the Riemannian heat kernel and change of variables show that $I_t$ is a class function on $T$. For $U\le T$, the heat kernel on a finite isometric quotient and integration over a fundamental domain give the displayed heat trace formula. Grouping by conjugacy classes proves equality for Gassmann equivalent subgroups. More generally the same argument allows fixed points of $T$ outside the freely acting subgroups; only the quotient under consideration must be smooth.

A finite isometry group decomposes $L^2(N)$ into its irreducible representation factors $V_{\rho }$ and multiplicity Hilbert spaces ${\mathcal{H}}_{\rho }$. The Laplacian commutes with the group, so it acts as the identity on each representation factor. The averaging projection gives $\dim V_{\rho }^{U_i}=|U_i{|}^{-1}{\sum }_{u\in U_i}{\chi }_{\rho }(u)$. For Gassmann equivalent subgroups these dimensions agree. Choosing unitary identifications of the invariant finite-dimensional factors therefore intertwines the quotient Laplacians, even when a noncompact covering manifold has no finite heat trace.

Let two finite locally isometric covers carry uniquely length-identified primitive closed geodesics ${\alpha }_i,{\beta }_i$. If their geometric intersection numbers differ, the covering surfaces are not isometric, because an isometry preserves both lengths and intersections. For base simple curves intersecting once, lift components correspond to cycles of their monodromy permutations. Their intersection number is the number of sheets common to the two cycles. This distinguishes covers even when every individual permutation has the same cycle type in both actions, as in Gassmann equivalence.

In a finite hyperbolic covering, primitive lifts of a primitive closed geodesic correspond to cycles of its monodromy permutation; an $r$-cycle gives length $r\ell (\gamma )$. In a real-analytic family of compact hyperbolic surfaces or hyperbolic cone orbifolds, a prescribed multiple of a simple curve's length can be isolated generically from unrelated lifted lengths. To exclude an identity of length functions, pinch the simple curve: only its peripheral powers can have length tending to zero, while intersecting curves are long by the collar lemma. The countably many remaining length equalities have empty interior, so the Baire category theorem supplies a parameter avoiding them. Only finitely many closed geodesics lie below a fixed local length bound; thus the isolation persists in an open parameter neighbourhood. This makes cycle-intersection information an intrinsic obstruction to an isometry, rather than an invariant of a chosen covering map alone.

If two simple base geodesics meet once, their lift components correspond to the cycles $C,D$ of their monodromy permutations. A lifted crossing belongs to the sheet labels common to both cycles, giving the displayed geometric intersection count. When their relevant lifted lengths identify these components independently of the covering map, different multisets of intersection counts obstruct an isometry between the covering surfaces.

Take a hyperbolic torus with a single cone point of angle $2\pi /7$. Its orbifold fundamental group is $⟨a,d\mid [d,a{]}^7=1⟩$. An epimorphism to $\mathrm{PSL}(3,2)$ whose commutator has exact order $7$ has torsion-free kernel. The point and line stabilizers have index $7$ and order $24$, hence avoid all nontrivial cone stabilizers. Their quotient covers are smooth surfaces with Euler characteristic $7(-6/7)=-6$, so have genus $4$. Their equal coset permutation characters give equal heat traces by the Sunada theorem. Nonisometry still requires a geometric distinction, such as the intersection test for nonisometric finite covers.

A Lambert quadrilateral with acute angle $\pi /14$ and sides $x,y$ incident to the opposite right-angle vertex satisfies the displayed relation. Reflect four copies around that vertex to make a symmetric hyperbolic quadrilateral whose four corner angles are $\pi /14$. Identifying opposite sides yields a torus with one cone angle $2\pi /7$. The two central axis loops have lengths $2x,2y$ and meet once at right angles. Cutting along one loop gives a pair of pants with two equal geodesic boundaries and the cone point; reglue the boundaries with a Fenchel–Nielsen twist to obtain the two-parameter cone-torus family. Its seven-sheeted torsion-free covers have genus four.

Let a finite group $T$ be generated by elements of exact orders $p,q$ whose product has exact order $r$, with $p^{-1}+q^{-1}+r^{-1}<1$. The epimorphism from the hyperbolic triangle group sending its generators to these elements has torsion-free kernel. Its quotient of the hyperbolic plane is a closed surface with a $T$ action. A subgroup acts freely precisely when it meets no conjugate of a nonidentity power of the three elliptic generators. Free almost conjugate subgroups give isospectral manifolds by the Sunada theorem.

Write triangle-group generators as products $a=s_1{s}_2$, $b=s_2{s}_3$ of adjacent side reflections. Conjugation by $s_2$ sends both to their inverses. A sheet relabelling satisfying the displayed two permutation identities therefore lifts this base reflection to an orientation-reversing isometry between the two hyperbolic covers. Thus nonconjugate subgroups in the orientation-preserving finite group can still give isometric covers.

For a finite isometric group action on a closed Riemannian surface, choose a disk in the free locus whose translates are disjoint. Modify the Riemannian metric there, and identically on its translates, by small supported changes with three distinguished high Gaussian curvature maxima. Give their heights distinct values occurring nowhere else, arrange them as a tight noncollinear cluster, and keep separate clusters far apart. Such markers can be made by local conformal bump functions controlling curvature and its derivatives. Every quotient isometry must send an entire labelled cluster to another. After identifying the marked disks with the original one, an isometry fixes its three noncollinear points and is therefore the identity locally. If $p_i:M\to U_i\backslash M$ are free finite quotients, any quotient isometry $F$ consequently makes $F\circ p_1=p_2\circ t$ on an open disk for some element $t$ of the original group. Two local isometries agreeing on an open set agree on the connected surface, by propagation along geodesics. The equality therefore holds globally and implies $t{U}_1{t}^{-1}\subseteq U_2$, hence equality for equal subgroup orders. Nonconjugate equal-order subgroups thus have nonisometric quotients for this marked invariant metric; constant curvature is not required.

Suppose a fixed pair of Gassmann equivalent finite covers of a compact two-dimensional orbifold has isometric limiting hyperbolic metrics. Choose nearby orbifold-smooth nowhere locally homogeneous metrics and rescale them to the displayed distinct areas. Scaling by a positive constant multiplies area by that constant in dimension two, preserves local rigidity, and preserves convergence when the constants tend to one. Sunada theorem supplies equal spectra within each pair. Local rigidity forces any quotient isometry to descend to the identity on the base; nonconjugate covering subgroups therefore exclude within-pair isometry. Distinct areas exclude all cross-pair isometries and spectra, because the leading heat trace coefficient is area divided by $4\pi t$. Fixed markings of the two isometric limiting covers give convergence to the same metric. The existence of those isometric limits is an additional geometric input, not a consequence of Gassmann equivalence alone.

Choose distinct odd primes and take products of the nonisomorphic Gassmann equivalent regular subgroups, selecting one of the two groups at each prime. Product conjugacy-class counts give $2^n$ Gassmann equivalent subgroups of the same finite product of symmetric groups. Their fundamental groups are distinguished by whether the unique Sylow subgroup at each chosen prime is abelian. The closed-manifold realization of a finitely presented fundamental group and Sunada theorem therefore produce $2^n$ pairwise nonhomeomorphic, and hence nonisometric, isospectral closed four-manifolds. Gassmann equivalence alone, without a topology distinction, does not imply nonhomeomorphism.

For two assemblies of congruent Euclidean tiles, let $M_s,N_s$ encode the gluing or boundary reflection at each labelled face. An invertible constant matrix $C$ satisfying $C{M}_s=N_{s}C$ carries tile restrictions of Laplacian eigenfunctions bijectively to those on the second assembly. Boundary values satisfy $M_{s}u=u$ and outward normal derivatives satisfy $M_{s}v=-v$, so the intertwining identities preserve matching and boundary conditions. Use diagonal $-1$ for a Dirichlet boundary condition and $+1$ for a Neumann boundary condition.

Suppose an invertible real matrix $C$ intertwines symmetric orthogonal tile-reflection matrices: $C{M}_s=N_{s}C$. Then $C^{T}C$ commutes with every $M_s$, so its positive inverse square root does too. The polar decomposition of an invertible real matrix therefore gives an orthogonal matrix $O=C(C^{T}C{)}^{-1/2}$ satisfying $O{M}_s=N_{s}O$. On a disjoint union of congruent tiles, $O$ preserves both the $L^2$ inner product and Dirichlet energy. The intertwining identities preserve the face trace conditions defining the assembled Sobolev space, so $O$ gives a unitary equivalence of the assembled Laplacians.

Take seven copies of a scalene triangle, with sides labelled $a,b,c$ opposite its three vertices, and reflect adjacent copies across their glued side. For the first planar domain, the glued pairs are $a:(2,3),(6,7)$, $b:(4,6),(5,7)$, $c:(1,5),(3,7)$; for the second they are $a:(1,3),(5,7)$, $b:(2,6),(3,7)$, $c:(4,5),(6,7)$. Both coloured trees have central tile $7$. For triangles sufficiently close to equilateral the interiors do not overlap. If its three angles are distinct and between $\pi /4$ and $\pi /2$, the only reentrant vertices are the original central vertices, with angles four times the corresponding triangle angles. They force any proposed isometry to preserve the labelled central triangle, which is incompatible with the different attachments on its arms.

For the spectral proof index the nonzero vectors and covectors of ${\mathbb{F}}_2^3$ by $1,\dots ,7$, using binary coordinates. The Fano plane incidence matrix is $B_{y,x}=1_{y\cdot x=0}$. It satisfies $B^{T}B=2I+J$, so is invertible, and intertwines the point and line permutations of each of the three elementary transvections. Put $D=\mathrm{diag}(1,1,-1,1,-1,-1,1)$. The signed reflection matrices for the Dirichlet boundary condition are $-D{P}_{s}D$ and $-D{Q}_{s}D$, hence $C=DBD$ intertwines them. Its polar orthogonal factor gives a unitary equivalence of the Dirichlet Laplacians, preserving both the $L^2$ inner product and the Dirichlet energy of the piecewise functions. This produces a three-parameter family of noncongruent isospectral planar domains.

Take four congruent octagonal regular polygons with four alternate sides labelled $a,b,c,d$. The intervening sides have Neumann boundary conditions. Pair equal face parameters in the positive reference boundary direction. The complete gluing data areAll unpaired labelled faces not in a Dirichlet column also have Neumann boundary conditions. Encode these rules by signed permutation matrices, using a positive swap for a paired interface and diagonal $-1$ or $+1$ for an unpaired Dirichlet boundary condition or Neumann boundary condition. The four-by-four Hadamard matrix $C$ above satisfies $C{M}_s=N_{s}C$ for every label. Explicitly, if its columns are $h_0,h_1,h_2,h_3$, applying $N_a,N_b,N_c,N_d$ gives respectively $(h_0,h_1,h_3,h_2)$, $(h_0,h_2,h_1,h_3)$, $(h_1,h_0,-h_2,-h_3)$, and $(h_0,-h_1,h_2,-h_3)$, which are exactly the columns of the corresponding $C{M}_s$. On every intervening face the face matrix is the identity.

Since $C^{T}C=4I$, the orthogonal matrix $C/2$ preserves the $L^2$ inner product and the sum of Dirichlet energies. Its intertwining relations map matching Sobolev traces and zero Dirichlet traces bijectively to those of the other assembly. It therefore preserves the quadratic form domains and the forms themselves. The associated self-adjoint operators are unitarily equivalent, proving equality of all eigenvalues with their multiplicities for the specified mixed boundary conditions.

The topology of disk tiles glued along disjoint boundary arcs shows that $D$, whose gluing graph is $0,1,2,3$ in a path, is a closed disc and is simply connected and orientable. The gluing graph of $B$ has edges $13,12,02,01,23$; it is connected with two independent cycles, so $\chi (B)=-1$ and ${\pi }_1(B)\cong F_2$. Its odd cycle $1,2,3,1$ makes it a nonorientable surface. Each assembled surface has nonempty polygonal boundary. This explicitly realizes isospectrality between a simply-connected orientable bordered surface and a nonorientable, non-simply-connected one. The boundary corners and mixed boundary conditions are essential qualifications; it does not contradict the Euler characteristic from the heat trace of a bordered surface for smooth boundaries and pure boundary conditions.

Take eight congruent hexagonal tiles with three pairwise nonadjacent distinguished faces. On tile indices modulo eight let $\sigma (j)=1-j$, $t(j)=-j$, and use $u_1(j)=3j$ for one assembly and $u_2(j)=3j+4$ for the other. Pair faces by reflection and leave fixed faces, and the three remaining faces, with Neumann boundary conditions. If $R{e}_j=e_{j+1}$, the displayed matrix intertwines all three face actions: its circulant coefficients satisfy $c_r=c_{-r}=c_{3r+4}$. Its eigenvalues are $1+(-1{)}^k+4\cos (\pi k/4)$, which are nonzero for $0\le k\le 7$. The transplantation theorem gives equal spectra. The second gluing graph is bipartite, with parts $\{0,5,6,7\}$ and $\{1,2,3,4\}$, so it is orientable. The first has the odd cycle $1,3,6,2,7,1$ and is nonorientable. Both graphs are connected with eight vertices and ten edges, and their bordered surfaces retract up to homotopy to the graphs. Both have Euler characteristic $-2$ and fundamental groups free of rank three. Neither is simply connected.

Consider assemblies of congruent polygonal tiles with same-labelled faces paired by reflection and pure Neumann boundary conditions on unpaired faces. An invertible intertwiner for all face permutation matrices preserves the Euler characteristic. The number of tiles is the dimension of the representation. For each face label the number of edge classes is $(n+\mathrm{tr}{M}_s)/2$. For each vertex label the number of vertex classes is the number of orbits of the subgroup generated by incident-face involutions, hence the dimension of its invariant vectors. Intertwining preserves all these numbers. Summing the vertex and edge counts proves equal $V-E+F$, including boundary corners. This argument uses genuine permutation matrices; signed matrices encoding mixed boundary conditions do not give the same topological count.

Reflection across a cut splits Laplacian eigenfunctions into odd and even parts. The odd restriction has a Dirichlet boundary condition on the cut; the even restriction has a Neumann boundary condition there. The displayed matrix intertwines the two-tile swap with $\mathrm{diag}(-1,1)$. A connected uniform-Dirichlet rectangle of width $2a$ is therefore isospectral to the disjoint union of width-$a$ rectangles with all-Dirichlet and three-Dirichlet/one-Neumann conditions. This illustrates why changing boundary conditions can conceal connectedness in the spectrum.

A propeller domain consists of seven congruent reflected triangles: one central triangle and three arms of two triangles each. There are different coloured gluing patterns whose Dirichlet Laplacian and Neumann Laplacian spectra agree by the transplantation theorem. Varying the three side lengths of a scalene triangle in a suitable open region gives three parameters; the central triangle is identifiable from the three reflex corners, allowing a direct nonisometry proof.

A class of metrics is spectrally rigid if equality of their specified spectra forces the metrics to be isometric, or, in a deformation version, if continuous isospectral deformations are trivial. The spectrum of a flat torus determines every two-dimensional flat torus up to isometry. The Wolpert generic spectral rigidity theorem is a generic, rather than universal, uniqueness statement.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.

A Riemannian manifold is a pair $(M,g)$ consisting of a smooth manifold $M$ and a Riemannian metric $g$. For each point $p$, the positive-definite inner product $g_p$ on its tangent space tells us the size and angle of possible instantaneous movements. The smooth manifold describes which local coordinates are possible; the Riemannian metric adds a choice of measurement.

For example, the plane can carry $d{s}^2=d{x}^2+d{y}^2$ or $d{s}^2=4d{x}^2+d{y}^2$. In the second Riemannian metric, a horizontal displacement costs twice as much length as the same coordinate displacement in the first. The underlying smooth manifold is unchanged. The coordinate transformation $X=2x,Y=y$ identifies the second geometry with an ordinary Euclidean plane, illustrating why coordinate appearance alone does not determine curvature.

A round sphere with its induced metric is another Riemannian manifold. Its tangent spaces each have an ordinary Euclidean inner product, although its global geometry is different from a plane's. This is the useful tension: measurements are Euclidean to first order at a point, while how those measurements fit together produces curvature and global restrictions on geodesics.

A linear isometry between tangent spaces of complete simply connected Riemannian manifolds extends to a global isometry when parallel transport along corresponding broken geodesics identifies their Riemann curvature tensors. In particular, complete simply connected manifolds of the same dimension and constant sectional curvature are isometric: their curvature tensors are determined by that common constant and their Riemannian metrics. This is the constant-curvature form often called Cartan's theorem.

For a connected complete Riemannian manifold of dimension $n\ge 2$ and constant sectional curvature $k>0$, its complete simply connected universal cover is the round sphere of radius $k^{-1/2}$. This is the constant-curvature case of Cartan-Ambrose-Hicks theorem. Thus the manifold is a quotient by a finite group of isometries acting freely. In even dimension, the determinant embeds this group in $\{1,-1\}$: every determinant-one orthogonal transformation in odd ambient dimension has eigenvalue one and therefore a fixed point. The only possible nonidentity element is the antipodal involution, giving Real projective space. In odd dimensions, free cyclic scalar actions on complex unit spheres instead produce lens spaces.

A Riemannian manifold is locally symmetric when its Riemann curvature tensor is parallel for the Levi-Civita connection. Along a geodesic, parallel transport then conjugates the Jacobi curvature operator at the initial point to the operator at every later point. Thus its eigenvalues stay constant. The condition is local and does not assert that global point reflections extend to isometries of the whole manifold.

Diagonalize the self-adjoint Jacobi curvature operator at the initial point and transport its orthonormal eigenbasis parallel along the geodesic. The Jacobi field equation reduces to $j_i^{\prime \prime }+{\lambda }_i{j}_i=0$. For zero initial position and initial derivative $b_i$, the solution is $b_i\sin (\sqrt{{\lambda }_i}t)/\sqrt{{\lambda }_i}$ when ${\lambda }_i>0$, $b_{i}t$ when ${\lambda }_i=0$, and $b_i\sinh (\sqrt{-{\lambda }_i}t)/\sqrt{-{\lambda }_i}$ when ${\lambda }_i<0$. Only positive eigenvalues contribute conjugate points, at $t=k\pi /\sqrt{{\lambda }_i}$. At a common zero the multiplicity is the sum of the contributing eigenspace dimensions. The displayed convention uses the curvature operator positive on a positively curved plane; reversing the tensor convention also reverses its tensor formula.

The map $(x,r)\mapsto {\exp }_x(r{\nu }_{\mathrm{in}})$ gives a collar in which the metric has this form. Two metrics with the same induced boundary metric can be identified by their normal collars, fixing the boundary and matching full tangent-space metrics there. To extend the local identification globally, interpolate the metrics, form their smoothly varying collars, and extend the resulting collar isotopy velocity by a cutoff. The time-dependent vector field vanishes on the boundary and agrees with that velocity near it, so its global flow gives the desired boundary-fixing diffeomorphism.

A compact manifold with boundary is simple when the boundary is strictly convex and any two points are joined by a unique geodesic depending smoothly on the endpoints, without conjugate points. Geodesics exit in finite time. These properties make off-diagonal boundary distance derivatives and the geodesic scattering relation well behaved.

For an inward unit boundary vector, the geodesic scattering relation returns the next boundary point and its outgoing unit velocity. On outward vectors it can be extended using the first backward exit, giving an involution on the full boundary unit bundle, with tangential vectors fixed. Exit time is separate lens data; on a simple manifold it equals boundary distance between the endpoint pair.

With the full boundary metric fixed, first variation gives $d_x{d}_g(w)=-⟨v_{\mathrm{in}},w⟩$ and $d_y{d}_g(z)=⟨v_{\mathrm{out}},z⟩$ for tangential endpoint variations. These covectors recover the tangential parts of both unit velocities. Their normal components follow from unit length and the inward/outward sign, so equal boundary distances give equal scattering relations. Without a fixed boundary metric, the bundle identification must first be gauged by a boundary-fixing diffeomorphism.

The boundary distance is the intrinsic distance between boundary points. Its infinitesimal limit along boundary curves recovers the induced metric on the boundary tangent bundle. It does not directly fix the normal and mixed metric components, which can be changed by boundary-fixing diffeomorphisms.

A Riemannian manifold is homogeneous when its isometry group acts transitively on points. For example, a Lie group with a left-invariant metric is homogeneous under left translations. Homogeneity gives the same local metric-ball geometry at every point, which yields the completeness of homogeneous Riemannian manifolds. It does not imply two-point homogeneity: that stronger property also controls directions and pairs at equal distance.

A connected Riemannian manifold is two-point homogeneous if its isometry group is transitive on ordered pairs at each fixed Riemannian distance. The equal-distance condition is necessary because isometries preserve distance. Round spheres and Euclidean spaces are examples. The unit tangent transitivity characterizes two-point homogeneity lemma shows that this pair condition is equivalent to transitivity on the unit tangent bundle.

For a connected Riemannian manifold, two-point homogeneity is equivalent to the isometry group being transitive on the unit tangent bundle. One direction follows by taking short equal-length radial geodesic segments: the Gauss lemma identifies their distance, and injectivity of the exponential map identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The Hopf-Rinow theorem supplies minimizing geodesics for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the geodesic equation matches their other endpoints.

In a homogeneous Riemannian manifold, choose $r>0$ so that one closed metric ball $\overline{B}(o,r)$ is compact. Such a ball exists from local compactness and the metric topology. Every radius-$r$ ball is isometric to it. A Cauchy sequence eventually lies in one such compact ball, has a convergent subsequence, and therefore converges. This proves metric completeness; the Hopf-Rinow theorem gives geodesic completeness and minimizing geodesics between points.

A conical singularity is a failure of smoothness at the apex of a metric cone. If $\vartheta$ has period $P$, small circles have circumference-to-radius ratio $aP$. The two-dimensional apex is smooth precisely when this equals $2\pi$. Euclidean black-hole regularity condition applies this test to the imaginary-time and radial plane.

A Killing-Yano tensor is a differential form satisfying the displayed equation for the Levi-Civita connection. Its covariant derivative is totally antisymmetric. Contraction with the velocity of an affinely parametrized geodesic gives a form that is carried by parallel transport along that geodesic. This differs from a Killing tensor, which is symmetric.

For a Killing-Yano two-form, ${\nabla }_a{Y}_{bc}$ is a differential three-form. Along a geodesic with velocity $v$, the covector $w_c=Y_{bc}{v}^b$ is carried by parallel transport, since $v^a{v}^b{\nabla }_a{Y}_{bc}=0$.

The square is a rank-two Killing tensor. It is symmetric because it pairs the covectors $Y_{a\cdot }$ and $Y_{b\cdot }$ using the metric. Along every affinely parametrized geodesic, its contraction $K(v,v)$ is the squared norm of the carried by parallel transport covector $Y(v,\cdot )$. Metric compatibility makes that norm constant, proving the Killing tensor equation. On a Riemannian manifold this tensor is positive semidefinite.

A Killing tensor is a symmetric covariant tensor satisfying the displayed equation for the Levi-Civita connection. Its contraction with $r$ copies of a geodesic velocity is constant along every affinely parametrized geodesic. The converse follows because the resulting symmetric derivative is determined by its homogeneous polynomial on tangent vectors. Rank one recovers the Killing vector field after metric duality.

A symmetric two-tensor satisfying this equation gives a quadratic geodesic first integral. Examples include the Riemannian metric, symmetric products of metric-dual Killing vector fields, and the square of a Killing-Yano two-form.

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.

A spin manifold is an oriented Riemannian manifold whose orthonormal frame bundle lifts from $SO(n)$ to its double cover $\mathrm{Spin}(n)$. Such a lift defines a spinor bundle.

A chosen spin structure on a spin manifold defines the complex spinor bundle associated to the spin representation. In dimension $2m$ its vector bundle rank is $2^m$. Its chirality matrix is $\Gamma =i^{m}c(e^1)\cdots c(e^{2m})$ in the negative Clifford multiplication convention. It satisfies ${\Gamma }^2=1$ and anticommutes with each $c(e^i)$, giving $S=S^{+}\oplus S^{-}$ and an odd Dirac operator. In odd dimension there is no such canonical chiral splitting of the irreducible spinor bundle.

A spin structure is a consistent choice of lifting oriented orthonormal frame changes to their spin double cover, permitting globally defined spinor fields. For a spatial circle in a worldsheet, the two choices give periodic and antiperiodic fermions, hence the Ramond sector and Neveu–Schwarz sector. Local field equations do not choose the global spin structure.

A Spin-c structure on an oriented rank-$n$ real vector bundle is a lift of its oriented frame bundle to ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))/\{(1,1),(-1,-1)\}$. Its determinant line bundle is associated with $[s,z]\mapsto z^2$; its First Chern class reduces modulo two to the second Stiefel–Whitney class. When such structures exist, their isomorphism classes form an affine space over $H^2$; tensoring by a complex line bundle of first Chern class $h$ changes the determinant class by $2h$. On a closed oriented three-manifold they always exist. On a four-manifold, the determinant class is characteristic: $⟨c_1(\mathfrak{s}),v⟩\equiv v\cdot v(\mathrm{mod}2)$.

A spinor field is a section of a spinor bundle. Clifford multiplication by tangent vectors acts fiberwise on spinor fields.

The Dirac operator is the first-order elliptic operator obtained by composing the spin connection with Clifford multiplication. On an even-dimensional compact spin manifold it exchanges positive and negative chirality spinors.

For the oriented round unit two-sphere, positive and negative spinor bundles have degrees $-1,+1$. Its Dirac operator has $D^2=C+1/4$ on their homogeneous sections, where $C$ is the angular-momentum Casimir operator; the eigenvalues of $D$ are the nonzero integers. Therefore the chiral phase is a unitary between the two L2 chiral spaces, commuting with $SU(2)$. For a smooth function $f$, $[D,f]=c(df)$ is bounded. The sign-operator resolvent integral expresses its phase commutator as a norm-convergent integral of compact resolvent products bounded by a constant times $(1+t^2{)}^{-1}$. Thus it intertwines multiplication by every continuous function modulo compacts. Its full graded phase is self-adjoint; the chiral phase is the component relevant to nonzero compressed indices.

Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral Dirac operator phase. The compressed map has parametrix $P{V}^{\ast }P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their homogeneous line-bundle representations on the two-sphere are respectively $E_0\oplus E_2\oplus E_4\oplus \cdots$ and $E_2\oplus E_4\oplus \cdots$, so the equivariant Fredholm index is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.

For a Twisted Dirac operator on a closed even-dimensional spin manifold, the theorem identifies its analytic Fredholm index with the integral of the top-degree part of the A-hat form times the Chern character. This is one geometric specialization of the general theorem for elliptic differential operators, which expresses the analytic index in terms of the topological class of the principal symbol. The heat proof pairs all positive eigenvalues by the odd Dirac operator, leaving only the index in the supertrace; Getzler rescaling computes its local limit from curvature.

Getzler rescaling combines the change $x\mapsto \sqrt{t}x$ in geodesic normal coordinates with the Clifford algebra filtration, assigning degree one to a derivative and to a Clifford generator and degree minus one to a coordinate. The associated graded Clifford algebra is the exterior algebra. This is useful because the spinor supertrace annihilates every Clifford monomial except the top-degree one. Applied to the Lichnerowicz formula for a twisted Dirac operator, it retains a quadratic curvature model, whose heat kernel is computed by a Gaussian ansatz; all terms of smaller rescaling degree disappear from the local index limit. Thus the cancellations in a supersymmetric heat trace can be computed before solving the complete curved heat equation.

On a spin manifold, tensor the spinor bundle $S$ with a Hermitian vector bundle $W$ carrying a unitary connection. Composing the induced tensor-product connection on a vector bundle with Clifford multiplication defines the twisted Dirac operator. In even dimension it maps $S^{+}\otimes W$ to $S^{-}\otimes W$ and conversely. On a closed manifold its chiral restriction $D_W^{+}:H^1(S^{+}\otimes W)\to L^2(S^{-}\otimes W)$ is a Fredholm operator.