Sphere configurations and tangent variations

Let $r\ge 2$ and $n\ge 1$ be integers. We use one-based labels for Euclidean coordinates: for $x\in {\mathbb{R}}^m$, the label $x_k$ denotes the supplier coordinate $x(k-1)$ for $1\le k\le m$. For $u,v\in {\mathbb{R}}^r$, write $u\cdot v={\sum }_{k=1}^r{u}_k{v}_k$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_2=\sqrt{u\cdot u}$ for the Euclidean inner product and norm. The unit sphere is $S^{r-1}=\{u\in {\mathbb{R}}^r:\mathrm{\parallel }u{\mathrm{\parallel }}_2=1\}$. A sphere configuration is an ordered $n$-tuple $u=(u_1,\dots ,u_n)\in (S^{r-1}{)}^n$.

The product Euclidean inner product of two tuples $v,w\in ({\mathbb{R}}^r{)}^n$ is $⟨v,w⟩={\sum }_{i=1}^n{v}_i\cdot w_i$. A tangent variation at $u$ is a tuple $v\in ({\mathbb{R}}^r{)}^n$ satisfying $u_i\cdot v_i=0$ for every $i$. These tuples form the tangent space $T_u(S^{r-1}{)}^n$ of the configuration space, with the product inner product restricted to it.

Given a real symmetric matrix $B=(b_{ij}{)}_{1\le i,j\le n}$, so that $b_{ij}=b_{ji}$, a pair interaction energy on configurations is $E_B(u)={\sum }_{i,j=1}^n{b}_{ij}(1-u_i\cdot u_j)$. The matrix and the product metric are part of the specified model. For $r=2$, a circle phase ${\theta }_i\in [0,2\pi )$ represents $u_i=(\cos {\theta }_i,\sin {\theta }_i)$.

math:def-euclidean-inner-product

Identify $({\mathbb{R}}^r{)}^n$ with ${\mathbb{R}}^{rn}$ by placing the scalar $(u_i{)}_k$ at supplier coordinate $(i-1)r+(k-1)$. Use the product Euclidean norm $\mathrm{\parallel }h{\mathrm{\parallel }}_2^2={\sum }_{i=1}^n\mathrm{\parallel }{h}_i{\mathrm{\parallel }}_2^2$, and define $F:({\mathbb{R}}^r{)}^n\to {\mathbb{R}}^n$ by $F_i(u)=u_i\cdot u_i$, where $F_i$ is output coordinate $i-1$. For each $u$, the map $L_u(v)=(2u_i\cdot v_i{)}_{i=1}^n$ is linear by Euclidean bilinearity (The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ). Expansion gives $F(u+h)-F(u)-L_u(h)=(\mathrm{\parallel }{h}_i{\mathrm{\parallel }}_2^2{)}_{i=1}^n$, whose Euclidean norm is at most ${\sum }_i\mathrm{\parallel }{h}_i{\mathrm{\parallel }}_2^2=\mathrm{\parallel }h{\mathrm{\parallel }}_2^2$: squaring this inequality leaves only nonnegative cross terms. Thus, for every $\epsilon >0$, taking $\delta =\epsilon$ makes the remainder at most $\epsilon \mathrm{\parallel }h{\mathrm{\parallel }}_2$ whenever $\mathrm{\parallel }h{\mathrm{\parallel }}_2<\delta$. This proves $DF(u)=L_u$ by The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder on the open domain ${\mathbb{R}}^{rn}$.

The derivative is continuous in $u$. Indeed, Cauchy–Schwarz (Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation) gives $\mathrm{\parallel }(L_u-L_w)v{\mathrm{\parallel }}_2^2=4{\sum }_i\mathrm{\mid }(u_i-w_i)\cdot v_i{\mathrm{\mid }}^2\le 4{\sum }_i\mathrm{\parallel }{u}_i-w_i{\mathrm{\parallel }}_2^2\mathrm{\parallel }{v}_i{\mathrm{\parallel }}_2^2\le 4\mathrm{\parallel }u-w{\mathrm{\parallel }}_2^2\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Consequently $\mathrm{\parallel }(L_u-L_w)v{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }u-w{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$, uniformly for unit $v$. The same inequality with $w=0$ and the quadratic remainder bound give $\mathrm{\parallel }F(u+h)-F(u){\mathrm{\parallel }}_2\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }h{\mathrm{\parallel }}_2+\mathrm{\parallel }h{\mathrm{\parallel }}_2^2$, so $F$ is continuous. Substituting an increment supported at the scalar coordinate $(u_j{)}_k$ into the derivative formula gives $\mathrm{\partial }{F}_i\mathrm{/}\mathrm{\partial }(u_j{)}_k=2(u_i{)}_k$ when $i=j$, and zero otherwise. Each nonzero entry changes in absolute value by at most $2\mathrm{\parallel }u-w{\mathrm{\parallel }}_2$, hence these partial derivatives are continuous. Together with continuity of the components this verifies that $F$ is $C^1$ in the componentwise convention of Cᵏ Euclidean maps and diffeomorphisms.

The configuration space is $F^{-1}(1,\dots ,1)$. At any point of this fibre, $DF(u)$ is onto: for any $a\in {\mathbb{R}}^n$, choose $v_i=(a_i\mathrm{/}2)u_i$, giving $DF(u)v=a$. Thus $(1,\dots ,1)$ is a regular value, and the regular-level-set tangent definition (The tangent space to a regular level set) gives precisely the displayed orthogonality conditions.

The product inner product is the Euclidean inner product on ${\mathbb{R}}^{rn}$; its restriction to the tangent subspace is therefore an inner product in the sense of Real and complex inner product spaces, with the inner product linear in the first argument: bilinearity, symmetry and positive definiteness are preserved on a linear subspace. The real matrix entries are supplied by The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations, with its zero-based entry $(i-1,j-1)$ denoted here by $b_{ij}$ for $1\le i,j\le n$. The energy is a finite real sum, so it is well defined for every configuration. The circle parametrization is bijective on $[0,2\pi )$ by math:thm-sine-and-cosine-parametrize-the-unit-circle, independently in each coordinate.

No extension of a circle-specific kernel argument or synchronization conclusion to higher spheres is asserted.