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A non-equatorial latitude on the round sphere is preserved transitively by rotations about its axis but has nonzero geodesic curvature. Its two complementary spherical caps are not isometric.

If a closed oriented surface is cut along a curve into two isometric surfaces with boundary, their Euler characteristics and total Gaussian curvatures agree. Applying Gauss-Bonnet to both halves, whose induced boundary orientations are opposite, gives

Suppose a group of isometries of an oriented Riemannian surface preserves a connected regular curve and acts transitively on it. Then the magnitude of its geodesic curvature is constant. The isometries must preserve the ambient surface metric; preserving only a curve's image in Euclidean space ${\mathbb{R}}^3$ does not determine its geodesic curvature on an unrelated surface.

A surface isometry preserves the Levi-Civita connection, arc length, and the magnitude of tangential acceleration $D_{s}T$, which is $|{\kappa }_g|$ for a regular curve parametrized by arc length. Transitivity makes it constant. After fixing curve and surface orientations, signed geodesic curvature is continuous, so connectedness makes it identically zero or of one constant nonzero value. A homogeneous nongeodesic latitude shows that transitivity does not force a geodesic.

A coordinate $q_i$ is ignorable, or cyclic, when $\partial L/\partial q_i=0$. The Euler--Lagrange equation then gives ${\dot{p}}_i=0$, so its generalized momentum is conserved.

A generalized linear mixed model adds latent random effects to the linear predictor of a generalized linear model.

IRLS fits a generalized linear model by repeatedly solving a weighted least-squares approximation to its score equations.

Logistic regression models the conditional log odds as an affine function of predictors.

For fixed exposures $x_i\ge 0$, a Poisson exposure model takes independent observationsIt has mean and variance $\theta x_i$ and is therefore heteroscedastic when the exposures differ.

A Poisson regression models independent count responses byIts likelihood is ${\prod }_i{e}^{-{\mu }_i}{\mu }_i^{y_i}/y_i!$.

The canonical link identifies the linear predictor with the exponential family's natural parameter: $g(\mu )=\theta$, where $\mu =b^{\prime }(\theta )$.

For a generalized linear model with variance function $V$, the Pearson estimator is

Binomial regression models independent $Y_i\sim \mathrm{Bin}(m_i,p_i)$ and relates $p_i$ to a linear predictor. Logistic and probit regression use the logit and inverse-normal links respectively.

A link function connects the response mean to the linear predictor through $g({\mu }_i)={\eta }_i$.

The linear predictor of a generalized linear model is the linear combination ${\eta }_i=x_i^T\beta$ of the explanatory variables.

If $F(s)=c{e}^{-\chi s}$ with $\chi >0$, the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing $K(x,y)=-B(x)e^{-\chi y}$ reduces the integral equation to one algebraic equation for $B(x)$ and reconstructs a single soliton.

For an embedded submanifold $M\subseteq N$ of a Riemannian manifold, the vector-valued second fundamental form isHere $\overline{\nabla }$ is the ambient Levi-Civita connection and the superscript denotes projection onto the normal bundle. For a hypersurface in Euclidean space with unit normal $N$, the scalar form is $II(X,Y)=⟨A(X,Y),N⟩=⟨-dN(X),Y⟩$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.

Two disjoint simple closed curves on a sphere bound an annulus $A$. If both are geodesics, its boundary geodesic-curvature terms vanish, whereas $\chi (A)=0$. Gauss-Bonnet would givewhich is impossible when $K>0$ everywhere.

The total Gaussian curvature of a closed surface is two pi times its Euler characteristic.

For a standard ring torus, Gaussian curvature is positive outside, negative inside, and zero along the top and bottom circles.

A simple closed curve of length $L$ on the unit sphere, enclosing a smaller region of area $0<A\le 2\pi$, satisfiesEquality holds exactly for a circle on the sphere. For a sphere of radius $R$, the formula becomes $L^2\ge A(4\pi -A/R^2)$.

A spherical cap of polar angle $\theta$ has $A=2\pi (1-\cos \theta )$ and boundary length $L=2\pi \sin \theta$, which give equality by substitution. The constrained first variation explains these candidates: stationarity of boundary length under fixed area forces constant geodesic curvature. Such a connected regular curve on the sphere lies in a plane and is a circle. This identifies smooth minimizers; existence and regularity of a minimizer are additional ingredients of the full proof, not consequences of stationarity alone.