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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 $\ell (\ell +d-1)$.

The normalized angular average is over the unit sphere, with solid angle measure $d\Omega$. Rotational symmetry gives $⟨e^i{⟩}_{\Omega }=0$, $⟨e^i{e}^j{⟩}_{\Omega }={\delta }^{ij}/3$, and $⟨e^i{e}^j{e}^k{e}^{\ell }{⟩}_{\Omega }=({\delta }^{ij}{\delta }^{k\ell }+{\delta }^{ik}{\delta }^{j\ell }+{\delta }^{i\ell }{\delta }^{jk})/15$. The constants follow by contracting indices and using $e^i{e}_i=1$. These identities extract photon angular temperature moments. For normalized spherical harmonics, $\int Y_{\ell m}^{\ast }{Y}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}$; their angular average instead has an extra factor $1/(4\pi )$.

On a sphere centred at the origin, an antipodal pair is the unordered pair $\{x,-x\}$. The antipodal map exchanges its two points.

The oriented metric volume form on a two-dimensional surface. On the unit sphere with standard orientation, $\omega =\sin \theta d\theta \wedge d\phi$ and $\int \omega =4\pi$. Its pullback of a differential form under a map measures signed local area.

In an orientable irreducible three-manifold, an incompressible cut is fundamental group-injective by the Loop theorem. Lift a three-manifold hierarchy to the universal cover. The terminal ball pieces are contractible and their gluing surfaces have contractible universal covers. The incidence graph is a tree, by the van Kampen theorem normal form for successive cuts. Finite subtrees are contractible by successively attaching a terminal piece along a contractible intersection. Every sphere map has compact image contained in a finite subtree, so all higher homotopy groups of the universal cover vanish.

For a complete connected $n$-dimensional Riemannian manifold with $\mathrm{Ric}\ge (n-1)kg$, $k>0$, put $D=\pi /\sqrt{k}$ andHere ${\omega }_{n-1}$ is the area of the unit $(n-1)$-sphere, and $v_k(r)$ is the Riemannian volume of a radius-$r$ ball in the round curvature-$k$ sphere. The Bishop-Gromov inequality asserts that $V_p(r)/v_k(r)$ is nonincreasing for $0<r<D$, tends to one at zero, and is at most one. In geodesic polar coordinates, the radial volume density before the cut time is at most $s_k(t{)}^{n-1}$. The bound uses trace Ricci curvature; with normalized Ricci curvature it reads $\overline{\mathrm{Ric}}\ge kg$.

The Borsuk-Ulam theorem says that recording $n$ continuous real-valued measurements on an $n$-dimensional unit sphere must give identical records somewhere on opposite sides. The measurements can vary wildly; continuity alone forces a coincidence at an antipodal pair.

Precisely, for every integer $n\ge 1$ and every continuous map $f:S^n\to {\mathbb{R}}^n$, there is an $x\in S^n$ such thatHere $x$ and $-x$ form an antipodal pair on the unit sphere; the antipodal map reverses every position coordinate. The target is the Euclidean space ${\mathbb{R}}^n$. Both points have the same entire output, so every one of the $n$ measurements agrees at this common pair. See Hatcher, Algebraic Topology, Corollary 2B.7, p. 176.

Circle example. Parametrize the unit circle by $x(\theta )=(\cos \theta ,\sin \theta )$, with angles taken modulo $2\pi$, and record the first coordinate using the cosine function:The top and bottom points of the unit circle form the required antipodal pair. Their first coordinates agree, while their second coordinates are $1$ and $-1$.

Sphere example. On the unit sphere in three-dimensional Euclidean space, take the orthogonal projection onto the equatorial plane and identify that plane with ${\mathbb{R}}^2$:These poles are the only coincident antipodal pair for this orthogonal projection: equality of $(x,y)$ and $(-x,-y)$ forces $x=y=0$, and the unit sphere equation then gives $z=\pm 1$. The Borsuk-Ulam theorem guarantees an antipodal pair for every continuous map into ${\mathbb{R}}^2$, even when no convenient projection formula is available.

Equal output need not mean equal position coordinates. Keeping all three coordinates gives a continuous map $i:S^2\to {\mathbb{R}}^3$ defined by $i(x,y,z)=(x,y,z)$, the restriction of the identity map of ${\mathbb{R}}^3$. Its outputs at an antipodal pair are always distinct, since $u=-u$ would force $u=0$, which is outside the unit sphere. Thus replacing the target ${\mathbb{R}}^2$ by ${\mathbb{R}}^3$ destroys this guarantee.

Why the circle case follows from the intermediate value theorem. For an arbitrary continuous map $f:S^1\to \mathbb{R}$, write $F(\theta )=f(x(\theta ))$ using the unit circle parametrization above. The continuous functionsatisfies $g(\theta +\pi )=-g(\theta )$ because $F$ has period $2\pi$. If $g(0)=0$, the first antipodal pair already works. Otherwise $g(0)$ and $g(\pi )$ have opposite signs, and the intermediate value theorem supplies a zero between them. At that zero, the two antipodal values of $f$ agree. In the cosine function example, $g(\theta )=2\cos \theta$. This short argument handles only $n=1$. See Hatcher, Algebraic Topology, discussion of Theorem 1.10, p. 32.

An ideal weather interpretation. Model the surface of Earth as a sphere. At one fixed instant, suppose temperature $T$ and pressure $P$ are globally defined continuous scalar measurements on that sphere. Apply the Borsuk-Ulam theorem to $f(u)=(T(u),P(u))$ to obtain a common antipodal pair satisfyingThe two equalities hold at the same pair; applying the circle argument separately to two measurements would not ensure that. This is a conclusion about the ideal continuous model, with both measurements taken at the same instant, rather than an assertion that an exactly matching pair has been empirically measured. Matoušek gives this motivating interpretation and explicitly cautions that physical temperature need not be continuous.^[ref]

Equivalent formulations. A continuous map $h:S^n\to {\mathbb{R}}^n$ is odd when $h(-x)=-h(x)$, extending the symmetry of an odd function. The Borsuk-Ulam theorem equivalently says every such map has a zero. For a general $f$, the difference $h(x)=f(x)-f(-x)$ has this symmetry; conversely, equal antipodal outputs of an odd map must be zero. Another equivalent formulation is that no odd map between spheres $S^n\to S^{n-1}$ exists. Normalizing a nonvanishing odd map by its Euclidean norm would produce such an odd map between spheres. These formulations are useful starting points for algebraic topology; the general proof is beyond the circle argument here.^[ref]

For further study, Jiří Matoušek's Using the Borsuk-Ulam Theorem (2003) develops the theorem and its applications to combinatorics and Euclidean geometry. An early publication is Karol Borsuk's 1933 paper Drei Sätze über die n-dimensionale euklidische Sphäre.

The boundary subcomplex $\partial {\Delta }^n$ consists of all proper faces of the $n$-simplex. Its geometric realization of a simplicial complex is homeomorphic to the sphere $S^{n-1}$, while $|{\Delta }^n|$ is homeomorphic to the closed disk $D^n$.

For a finite group generated by $a,b$, take the regular covering of a thrice-punctured sphere with monodromy $a,b,(ab{)}^{-1}$ and fill each lifted puncture by a disk. Locally the projection is $z\mapsto z^m$, where $m$ is the order of its monodromy. The result is a closed oriented topological surface with an orientation-preserving group action. The nontrivial stabiliser subgroups are the conjugates of $⟨a⟩$, $⟨b⟩$ and $⟨ab⟩$ at the three types of ramification points. Counting lifted cells gives the displayed Euler characteristic. A subgroup acts freely exactly when it meets every conjugate of these three cyclic groups trivially. Its quotient is then a regular covering of degree equal to the subgroup order, and quotient Euler characteristic is divided by that order.

For three-dimensional Brownian motion, every radius $r>0$ and every initial point satisfyIf the initial point is on one of the spheres, it is already a hit at time zero. If it is inside a lattice-centered open ball, its eventual exit hits that sphere. Otherwise choose at each integer time a nearest lattice center. The current displacement from it has norm at most $\sqrt{3}/2$, so the multivariate normal distribution of the next unit increment gives a uniform positive chance of entering its radius-$r/2$ open ball. The geometric tail bound from a uniform escape probability makes eventual entry certain, and continuity forces a boundary crossing. This periodic-family recurrence does not contradict the Brownian sphere-hitting probability in dimension three for one fixed sphere.

The celestial sphere is the sphere of viewing directions around an observer; distances to objects are suppressed. Great circles define coordinate reference planes.

For a sphere with centre $c$ and radius $a$, and a plane $n\cdot r=p$ with unit normal $n$, the orthogonal projection of $c$ onto the plane is $b=c+(p-n\cdot c)n$. The orthogonal decomposition $r-c=(r-b)+(b-c)$ gives the intersection circle radius by the Pythagorean theorem. A genuine circle requires $|p-n\cdot c|<a$; equality gives tangency and larger distance gives no intersection. The signed displacement is not determined by the two radii alone: it is $\pm \sqrt{a^2-{\rho }^2}$ with sign fixed by the oriented plane.

For two independent uniform points on the surface of a sphere, rotational invariance permits fixing the first at the north pole. The surface area of a latitude band of angular width $d\theta$ is $2\pi R^2\sin \theta d\theta$, yielding the displayed density for their central angle. Equivalently their unit position vectors have a dot product uniformly distributed on $[-1,1]$. The law is independent of sphere radius.

A connected compact surface is determined by its orientability, number $b$ of boundary circles and genus. An orientable surface of genus $g$ has Euler characteristic $2-2g-b$; a surface which is not orientable with $r$ crosscaps has Euler characteristic $2-r-b$. In particular, a closed connected surface with vanishing first mod-two homology is a sphere, and a connected closed orientable surface of Euler characteristic zero is a torus.

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.

For a nonnegative integer $d$ and an abelian group $A$,The one-point compactification is the sphere $S^d$, and it is locally contractible at the added point. Apply the compact-support comparison with a one-point compactification and the reduced cohomology of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.

A symplectomorphism of a closed symplectic surface carries the complementary components of a separating simple closed curve to those of its image, preserving their unordered symplectic areas. On the unit sphere a latitude at height $h$ has complementary areas $2\pi (1-h)$ and $2\pi (1+h)$. Thus the equator cannot be sent to any nonequatorial latitude.