Sphere

A sphere is the set of points at one fixed positive Euclidean distance from a center.

A spherical shell is the region between two concentric spheres. A thin mass shell at radius $r$, thickness $dr$ and mass density $\rho (r)$ has mass $4\pi r^2\rho (r)dr$ to first order. Decomposing a spherical body into such shells makes the spherical shell theorem applicable to its gravitational potential energy.

The Cartesian product $S^p\times S^q$ has one cell in each of dimensions $0,p,q,p+q$, counting the two middle cells separately when $p=q$. For positive $p,q$, its integral cohomology ring has generators $a,b$ of degrees $p,q$, with $a^2=b^2=0$ and $ba=(-1{)}^{pq}ab$. The class $ab$ is a product orientation class. This follows from the Künneth theorem.

For positive $p,q$, quotient $S^p\times S^q$ by $(x,y)\mapsto (-x,-y)$. This is a free double covering space action. Its closed manifold quotient is orientable exactly when $p+q$ is even, because the product mapping degree is $(-1{)}^{p+q+2}$. If $p,q\ge 2$, the product is simply connected and the quotient has fundamental group $\mathbb{Z}/2$. For $p=q=2$, the quotient has integral cohomology groups $\mathbb{Z}$ in degrees $0,4$, $\mathbb{Z}/2$ in degrees $2,3$, and zero elsewhere. Every positive-degree cup product is zero. These conclusions follow from Poincare duality, the Euler characteristic under a finite covering, and the universal coefficient theorem for cohomology.

Surface area is the two-dimensional measure induced on the unit sphere in three dimensions. It is invariant under orthogonal rotations. It is not the restriction of ambient three-dimensional Lebesgue measure, under which the sphere has measure zero. Normalizing by the sphere area gives a probability measure.

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

For distinct points and $0<\lambda <1$, the distance-ratio locus is a sphere with center $(\mathbf{a}-{\lambda }^2\mathbf{b})/(1-{\lambda }^2)$ and radius $\lambda |\mathbf{a}-\mathbf{b}|/(1-{\lambda }^2)$. Completing the square proves this formula. It encloses $\mathbf{a}$ but excludes $\mathbf{b}$, because their distances to the center are $\lambda R$ and $R/\lambda$. It is the zero-potential locus of two opposite-sign point charges with magnitude ratio $\lambda$. Equal magnitudes instead give a plane.

A hemisphere is either half of a sphere cut by a plane through its center. Its curved surface has area $2\pi r^2$. Closing that surface with its equatorial disk encloses a half-ball of volume $2\pi r^3/3$. The curved surface and enclosed solid have distinct roles in surface-cost and volume calculations.

For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite open hemispheres are disjoint.

Choose $N$ points of $S^d$ such that every $d+1$ of their position vectors are linearly independent. Any equatorial hyperplane then contains at most $d$ points. If $N-d\ge 2r-1$, at least one of each opposite pair of open hemispheres contains $r$ points. Such configurations exist: take distinct real parameters $t_i$ and normalize $(1,t_i,\dots ,t_i^d)$; the Vandermonde determinant proves the required independence. This weaker one-of-two guarantee already suffices for the topological lower bound in Lovász theorem on Kneser graphs.

For integers $n=2k+d$ with $k\ge 1$ and $d\ge 0$, there are $n$ labelled points of $S^d$ such that every open hemisphere contains at least $k$ of them. The normalized vectors $(-1{)}^i(1,t_i,\dots ,t_i^d)$ for increasing $t_i$ give a signed moment curve construction. A nonzero polynomial of polynomial degree at most $d$ cannot have enough sign changes for fewer than $k$ positive alternating sample values. Zero sample values can be perturbed negatively by a Lagrange interpolation polynomial. Labels allow repeated positions when $d=0$.

A great circle is the intersection of a sphere with a plane through its centre. Its shorter arcs are geodesics of the round sphere; celestial meridians and the equator are examples.

For $N\ge 1$ great circles on a sphere, with no coincident circles or triple intersection, the number of open regions is $R_N=N(N-1)+2$. The first circle gives two regions. Each subsequent circle meets the previous $n-1$ circles at $2(n-1)$ distinct points and is divided into that many arcs, each splitting a previous region. Summing the increments proves the formula. If the circles are $a_i\cdot x=0$, each region has one fixed sign vector $(\mathrm{sgn}(a_i\cdot x){)}_i$. Conversely a realized strict sign vector has a connected region: its cone is an intersection of open half-spaces, and radial projection of line segments connects any two points.

The gnomonic projection intersects rays from a sphere’s centre with a tangent plane. If $\mathbf{b}$ is its unit normal and $\mathbf{e},\mathbf{n}$ are tangent basis vectors, a direction $\mathbf{s}$ maps to $f(\mathbf{s}\cdot \mathbf{e},\mathbf{s}\cdot \mathbf{n})/(\mathbf{s}\cdot \mathbf{b})$. It maps great circle arcs to straight lines and is singular at the tangent-plane horizon.

The unit sphere $S^n\subseteq {\mathbb{R}}^{n+1}$ is the regular level set $\Vert x{\Vert }^2=1$. Its tangent space is

The two-sphere is the two-dimensional sphere embedded in three-dimensional Euclidean space.

The three-sphere is the unit sphere in four-dimensional Euclidean space:Identifying ${\mathbb{R}}^4$ with ${\mathbb{C}}^2$ realizes it as $\{(z_1,z_2):|z_1{|}^2+|z_2{|}^2=1\}$.

A solid angle is the area cut out on a unit two-sphere by a cone of directions. Its SI unit is the steradian, and a full sphere subtends $4\pi$ steradians.

A steradian is the solid angle subtended by a patch of area $r^2$ on a sphere of radius $r$. The whole sphere subtends $4\pi \mathrm{sr}$.

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 $1/(4\pi )$.