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Writing $f=1-2M(v)/r$, the radial Christoffel symbols are ${\Gamma }^v{}_{vv}=M/r^2$, ${\Gamma }^r{}_{vv}=fM/r^2+M^{\prime }/r$ and ${\Gamma }^r{}_{vr}=-M/r^2$. The angular couplings are ${\Gamma }^v{}_{\theta \theta }=-r$, ${\Gamma }^r{}_{\theta \theta }=-fr$, with ${\sin }^2\theta$ factors for the corresponding $\varphi \varphi$ terms, ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\varphi }{}_{r\varphi }=1/r$, ${\Gamma }^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\Gamma }^{\varphi }{}_{\theta \varphi }=\cot \theta$. The lower indices are symmetric. The plus sign of $M^{\prime }/r$ follows from the ingoing $+2dvdr$ cross term.

For a nondegenerate metric tensor and its Levi-Civita connection, contracting the Christoffel symbol formula gives ${\Gamma }^{\rho }{}_{\rho \mu }=\frac{1}{2}{g}^{ab}{\partial }_{\mu }{g}_{ab}$. The Jacobi determinant derivative formula identifies this with ${\partial }_{\mu }\log \sqrt{|\det g|}$. The volume density, rather than the determinant without an absolute value, works in every fixed metric signature.

The difference of two affine connections is a smooth tensor field of type $(1,2)$. Linearity over smooth functions holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta (X,fY)=f\Delta (X,Y)$. This tensoriality makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the Christoffel symbols, although either collection of connection coefficients individually is not a tensor.

The Euclidean metric is the constant Riemannian metric on ${\mathbb{R}}^n$ given in Cartesian coordinates by $g={\sum }_i(d{x}^i{)}^2$. Its Levi-Civita connection has zero Christoffel symbols in those coordinates, so it is flat and its geodesics are straight lines. The Euclidean norm is the pointwise length determined by this metric.

For smooth stationary fields on ${\mathbb{R}}^3$, set ${\Gamma }^i{}_{tt}=-E^i$ and ${\Gamma }^i{}_{tj}={\Gamma }^i{}_{jt}={ϵ}^i{}_{jk}{B}^k$, with all other Christoffel symbols zero. This torsion-free connection makes the equation $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$ a geodesic equation with time as affine parameter. Arbitrary smooth fields suffice; a metric realization imposes additional potential conditions.

For smooth spatial $\varphi$ and $g(c)=-c^2{e}^{2\varphi /c^2}d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, the only nonzero Christoffel symbols are ${\Gamma }^t{}_{ti}={\Gamma }^t{}_{it}=c^{-2}{\partial }_i\varphi$ and ${\Gamma }^i{}_{tt}=e^{2\varphi /c^2}{\partial }_i\varphi$. Consequently the Levi-Civita connection converges smoothly on compact sets as $c\to \infty$ although the metric itself diverges in these coordinates. The limiting torsion-free connection has only ${\Gamma }^i{}_{tt}={\partial }_i\varphi$ and gives $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$ when $t$ is an affine parameter. With $R^a{}_{bcd}={\partial }_c{\Gamma }^a{}_{db}-{\partial }_d{\Gamma }^a{}_{cb}+\cdots$, its Ricci tensor is $(\Delta \varphi )dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}={\partial }_j{\partial }_i\varphi$. Thus zero Ricci tensor is the Laplace equation, while full flatness would require the entire Hessian to vanish.