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A pregeodesic is a curve whose tangent obeys ${\nabla }_{U}U=\kappa U$ for some scalar $\kappa$. A change of parameter makes it a geodesic with an affine parameter. A varying speed along a geodesic changes its parameter, not its image.

The symmetric covariant two-tensor defined using the Levi-Civita connection by ${\mathrm{Hess}}_{g}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Its negative metric trace is the positive Laplace-Beltrami operator applied to $f$. Along a geodesic it gives the second derivative of $f$.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.

In curvature $-1$, its length is $2\mathrm{arcosh}3$. The level-two principal congruence subgroup has traces congruent to two modulo four, so the smallest hyperbolic absolute trace is six, achieved by $\left(\begin{array}{cc}5& 2\\ 2& 1\end{array}\right)$. The hyperbolic translation length formula gives the result; the shortest closed geodesic is nonsimple.

The square is a rank-two Killing tensor. It is symmetric because it pairs the covectors $Y_{a\cdot }$ and $Y_{b\cdot }$ using the metric. Along every affinely parametrized geodesic, its contraction $K(v,v)$ is the squared norm of the carried by parallel transport covector $Y(v,\cdot )$. Metric compatibility makes that norm constant, proving the Killing tensor equation. On a Riemannian manifold this tensor is positive semidefinite.

The difference of two torsion-free connections is symmetric, because subtracting their zero-torsion identities gives $\Delta (X,Y)=\Delta (Y,X)$. Agreement of parametrized geodesics makes this difference skew-symmetric as well, by parametrized geodesics determine the symmetric part of an affine connection. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized geodesics.

For a nondegenerate induced metric tensor, a totally geodesic submanifold has vanishing second fundamental form; equivalently, every ambient geodesic initially tangent to it remains in it. This extends the notion of a totally geodesic hypersurface to arbitrary codimension.

For a connected Riemannian manifold, two-point homogeneity is equivalent to the isometry group being transitive on the unit tangent bundle. One direction follows by taking short equal-length radial geodesic segments: the Gauss lemma identifies their distance, and injectivity of the exponential map identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The Hopf-Rinow theorem supplies minimizing geodesics for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the geodesic equation matches their other endpoints.

A regular curve with tangent $v=dx/du$ is an unparametrized geodesic of an affine connection if ${\nabla }_{v}v=f(u)v$. Choose a new parameter $s$ with $ds/du=r(u)>0$ and $r^{\prime }/r=f$. Then $w=dx/ds=v/r$ obeys ${\nabla }_{w}w=r^{-2}({\nabla }_{v}v-(r^{\prime }/r)v)=0$. Conversely a reparametrized affine geodesic has acceleration proportional to its tangent. This criterion concerns the geometric curve, not a fixed parametrization.