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For positive coordinate charts put $J=\partial y/\partial x$. The metric tensor transforms by $g_x=J^T{g}_{y}J$, giving $\det g_x=(\det J{)}^2\det g_y$. The top wedge product of differential forms transforms by $d{y}^1\wedge \cdots \wedge d{y}^n=(\det J)d{x}^1\wedge \cdots \wedge d{x}^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global Riemannian volume form.

The total Riemannian volume is the integral of the Riemannian volume form, or its density on a nonorientable manifold. In local coordinates the density is $\sqrt{\det g}d{x}^1\cdots d{x}^d$. It is preserved by Riemannian isometries. On a closed manifold it is the leading coefficient among the integrated heat invariants, and on a surface it is the surface area.

For a surjective Riemannian submersion $\pi :M\to N$, let $H={\sum }_{\alpha }({\nabla }_{V_{\alpha }}{V}_{\alpha }{)}^{\mathrm{hor}}$ be the unnormalised mean curvature vector of its fibres, using an orthonormal vertical frame. With the positive Laplace-Beltrami operator, tracing the Riemannian Hessian in horizontal and vertical directions gives ${\Delta }_M(f\circ \pi )=({\Delta }_{N}f)\circ \pi +d(f\circ \pi )(H)$. Horizontal traces agree with those on the base; a vertical Hessian term is $-d(f\circ \pi )({\nabla }_{V_{\alpha }}{V}_{\alpha })$. Thus minimal fibres, in particular totally geodesic submanifolds as fibres, make pullback an injection from each base eigenspace into the corresponding total-space eigenspace. Its image is exactly the eigenfunctions constant on each whole fibre.

At any point of a Riemannian manifold, choose geodesic normal coordinates and orthonormalize their coordinate fields smoothly. At the point, the metric is the identity and its first derivatives vanish, so the orthonormalization coefficients have zero first derivatives. The Christoffel symbols also vanish there. This gives a local Riemannian orthonormal frame with the displayed property, useful for invariant identities evaluated at one point.

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.

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 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.

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 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.

The chordal metric on the Riemann sphere is $\chi (z,w)=|z-w|/\sqrt{(1+|z{|}^2)(1+|w{|}^2)}$ for finite points, with $\chi (z,\infty )=(1+|z{|}^2{)}^{-1/2}$. It is half the Euclidean chord distance between stereographic images on the unit sphere, so the triangle inequality follows from the Euclidean one. If $\rho$ is round spherical geodesic distance, then $\rho =2\arcsin \chi$ and $2\chi \le \rho \le \pi \chi$. These inequalities transfer uniform-continuity estimates between spherical and chordal distances, including at poles of meromorphic functions.

For a smooth compact complex surface, the degree-four part of its Todd class is $(c_1^2+c_2)/12$. Applying Riemann–Roch to its structure sheaf gives the formula. For a K3 surface, $c_1=0$ and $\chi ({\mathcal{O}}_X)=2$, so its topological Euler characteristic is $c_2=24$.

For a closed point $p$ on a smooth projective curve, the quotient in $0\to L(-p)\to L\to L{|}_p\to 0$ has $k$-dimension $[k(p):k]$. Additivity of Euler characteristic therefore changes $\chi (L)$ by that residue-field degree. Every line bundle is associated with a divisor, so iteration proves the displayed formula. The dualizing sheaf on a smooth projective curve identifies $h^1(L)$ with $h^0({\Omega }_C^1\otimes L^{-1})$ and converts this Euler-characteristic formula into the usual divisor form of Riemann-Roch theorem.

For finitely many distinct points $P_i$ on a smooth projective genus-$g$ curve, choose $N\ge 2g+1$. The Riemann-Roch theorem gives $\ell (N{P}_i)-\ell ((N-1)P_i)=1$, so choose $f_i$ with a sole pole of exact order $N$ at $P_i$. Then ${\sum }_i{f}_i$ has exactly those poles with that order: all other summands are regular at each $P_i$, so no principal part can cancel. This is an explicit construction rather than an unspecified generic choice of section.

An initially resting perfect gas with adiabatic index $\gamma >1$ next to a piston $X(t)=-f{t}^2/2$ develops a smooth rarefaction. The incoming invariant gives $c_p=c_0-(\gamma -1)ft/2$. Pressure falls as $p_p=p_0(1-t/t_v{)}^{2\gamma /(\gamma -1)}$ until $t_v$, when the piston speed is $2c_0/(\gamma -1)$. Thereafter the gas edge moves at that limiting constant speed while the piston accelerates away, opening a vacuum gap.

A proper Riemann integral on a compact interval requires a bounded integrand, but a derivative can be unbounded. For $f(x)=x^2\sin (1/x^2)$ away from zero and $f(0)=0$, the difference quotient gives $f^{\prime }(0)=0$, whereas $f^{\prime }(x)=2x\sin (1/x^2)-2\cos (1/x^2)/x$ for $x\ne 0$. At $x_n=(2\pi n{)}^{-1/2}$, this derivative is $-2/x_n$ and is unbounded. The fundamental theorem of calculus version that integrates a derivative therefore needs an explicit integrability hypothesis.

Let $|f|\le M$ on $[0,1]$. Suppose finitely many nonoverlapping closed intervals, of total length at least $1-\delta$, can be chosen for each $\delta >0$, with $f$ Riemann integrable on each. Refine the intervals to make their total Darboux sum gap less than $\epsilon /2$. The remaining length contributes at most $2M\delta$. Taking $\delta <\epsilon /(4M)$ proves global Riemann integrability by the Riemann integrability criterion.

Let $K$ consist of numbers whose decimal digits all lie in a fixed set $D\subseteq \{1,\dots ,8\}$ with $m<10$ digits. The $m^N$ allowed prefixes give closed decimal intervals of length $10^{-N}$ covering $K$, with no endpoints in $K$. A partition of an interval at those endpoints gives an upper Darboux sum at most $(m/10{)}^N$. Every cell contains a terminating decimal outside $K$, so the lower Darboux sum is zero. Thus the indicator function is Riemann integrable with Riemann integral zero. Such sets illustrate how an uncountable set may have an integrable indicator function of zero integral.

Let $f$ be bounded on $[a,b]$ and continuous on $(a,b]$. For any positive error tolerance, isolate $[a,a+\delta ]$. Its contribution to the difference of upper Darboux sums and lower Darboux sums is at most $2\Vert f{\Vert }_{\infty }\delta$. On $[a+\delta ,b]$, uniform continuity supplies a partition of an interval with arbitrarily small remaining oscillation sum. The Riemann integrability criterion then proves integrability, even when the values oscillate without a limit at the endpoint.

For a common interval partition, the oscillation of a function on an interval satisfies ${\mathrm{osc}}_I(f+g)\le {\mathrm{osc}}_{I}f+{\mathrm{osc}}_{I}g$. Refining two good Darboux sums therefore proves integrability of the sum. Lipschitz composition preserves Riemann integrability, so absolute values preserve it too. The identity $\max (f,g)=(f+g+|f-g|)/2$ then proves the assertion for the maximum.

The indicator of the rational numbers is not Riemann integrable on a nondegenerate interval: every subinterval contains both rational and irrational points, so its lower sum is zero and its upper sum is the interval length.