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WriteThe Levi-Civita connection ofisThe assumed first-derivative condition makes ${\Gamma }_{\alpha \beta }^{\rho }(p)=0$. Differentiating once more gives, at $p$,Its contractions areSubstitution into the defining expression for $C_{\alpha \beta \gamma \delta }$ cancels every Hessian term, leavingSince $p$ was arbitrary, this is the Weyl tensor of a conformally flat metric.

Locally write the null hypersurface as a level set $u=0$ and its normal as $n_a={\nabla }_{a}u$. Since a null normal is also tangent, $n^a$ lies within the hypersurface. The symmetry of the Levi-Civita connection givesThe scalar $n^2$ vanishes on the hypersurface, so its gradient there is normal and hence proportional to $n_a$:This is the nonaffine geodesic equation. A rescaling of $n$ removes $\kappa$, proving that the null hypersurface normal generates null geodesics lying in the hypersurface.

For coordinates $x^{\alpha }$, the Christoffel symbols of the Levi-Civita connection arewhere $g^{\alpha \delta }$ is the inverse metric.

For the two-dimensional hyperbolic metric in polar coordinates,Only $g_{\varphi \varphi }$ has a nonzero derivative, namelyIt follows that the nonzero connection coefficients are

For local coordinates $x^1,\dots ,x^n$, a curve is an affinely parametrized geodesic exactly when it satisfies the geodesic equationwhere the Christoffel symbols of the Levi-Civita connection are

Write ${\dot{x}}^{\mu }=d{x}^{\mu }/ds$ and $L=\sqrt{-g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }}$. Varying $T=\int Lds$ with fixed endpoints and then choosing proper time as parameter, for which $L=1$, gives the Euler-Lagrange equationExpanding the derivative, raising the free index with the inverse metric, and using the symmetry of ${\dot{x}}^{\alpha }{\dot{x}}^{\beta }$ gives the geodesic equationThese are the Christoffel symbols of the Levi-Civita connection.

For any vector field $Y$, apply the Leibniz rule to the scalar $\omega (Y)$:Using ${\mathcal{L}}_{V}Y=[V,Y]$ and expanding the Lie bracket in a coordinate chart leavesThe connection terms cancel because the Levi-Civita connection is torsion-free. Applying this formula to each slot of the metric tensor and using metric compatibility givesEquivalently, one may prove both identities at a point in normal coordinates; since both sides are tensors, the result then holds in every coordinate system.

The Levi-Civita connection isFor $g_{\alpha \beta }={\eta }_{\alpha \beta }+h_{\alpha \beta }$, retain only first-order terms in $h$. Substitution into the curvature formula and contraction gives the linearized Ricci tensor and scalarHence

Affinely parametrized geodesics are the critical curves of the geodesic LagrangianFor $w=1+r^2$ and $w^{\prime }=2r$, the nonzero Christoffel symbols, up to symmetry in the lower indices, areThey follow either from the Euler-Lagrange equations or directly from the Levi-Civita connection formula.

Because the Levi-Civita connection preserves the spacetime volume form, only derivatives of $X$ remain in ${\mathcal{L}}_{X}T$. Lower the raised indices in $T_{ab}{}^{cd}={ϵ}_{ab}{}^{cd}$, contract successively with the supplied products of two Levi-Civita tensors, and separate ${\nabla }_a{X}_b$ into antisymmetric, trace and symmetric trace-free parts. The antisymmetric part cancels automatically, while ${\mathcal{L}}_{X}T=0$ is equivalent to vanishing of the symmetric trace-free part:Thereforeso $\boxed{\alpha =1/2}$. This is the four-dimensional Conformal Killing equation, and $X$ is a Conformal Killing vector field.

In the Levi-Civita connection, the background metric is constant and replacing $g^{\alpha \rho }$ by its $O(ϵ)$ correction would multiply a derivative of $h$ and produce an $O({ϵ}^2)$ term. Hence the linearized Levi-Civita connection isThe products of two connection coefficients in the Riemann curvature tensor are also quadratic and may be discarded. Lowering its first index with ${\eta }_{\mu \nu }$ gives

Because the coordinate change is ${\tilde{x}}^{\alpha }=x^{\alpha }-{\xi }^{\alpha }$, the perturbation changes at linear order bySubstitution into $R_{\mu \rho \alpha \beta }^{(1)}$ produces terms containing three partial derivatives of $\xi$. Since partial derivatives commute, every term cancels another with the opposite sign. ThereforeThis is the gauge invariance of the linearized Riemann tensor.

Yes to both questions. Since ${\omega }_g$ has top degree, $d{\omega }_g=0$. Moreover $\ast {\omega }_g=1$, so the formula for the codifferential givesThereforeand the Riemannian volume form is a harmonic differential form.

The Levi-Civita connection preserves both the Riemannian metric and its chosen orientation. At any point, extend a positively oriented orthonormal basis to a local frame whose covariant derivatives vanish at that point. Differentiating ${\omega }_g(e_1,\dots ,e_n)=1$ there gives $\nabla {\omega }_g=0$. Hence ${\omega }_g$ is a parallel differential form.

The flow of a Killing vector field consists of local isometries. Isometries preserve both the metric and its unique torsion-free metric-compatible Levi-Civita connection, hence ${\mathcal{L}}_K\nabla =0$. Set $X=K$ and $Y=Z=U$ in part (b). Since ${\nabla }_{U}U=0$,and antisymmetry of the first two curvature arguments givesThis is the geodesic deviation equation with connecting field $K$: applying the isometry to the original geodesic produces a neighboring geodesic, and curvature determines their relative acceleration.

For a spatial index $j$, the background Christoffel symbols giveTogether with the $3H\delta T^0{}_j$ trace term, the first two contributions combine to $2H\delta T^0{}_j$. Varying the Levi-Civita connection and contracting it with ${\bar{T}}^{\mu }{}_{\nu }=\mathrm{diag}(-\bar{\rho },\bar{P},\bar{P},\bar{P})$ givesand the surviving combination reduces toConsequently the spatial momentum equation is

For positive smooth $A$, this Lorentzian manifold has flat $(t,x,y)$ slices scaled by $A(z{)}^2$. Translation in each slice coordinate gives a Killing vector field. An orthonormal coframe is $e^0=Adt$, $e^1=Adx$, $e^2=Ady$, $e^3=dz$. Its Levi-Civita connection and Ricci tensor are controlled by $a=A^{\prime }/A$ and $q=A^{\prime \prime }/A=a^{\prime }+a^2$.

The product of Riemannian manifolds has metricThe tangent summands are orthogonal, and positivity follows because a nonzero product tangent vector has a nonzero component. Its Levi-Civita connection is the product affine connection of the two factor Levi-Civita connections: mixed torsion is zero and metric compatibility follows from independence of the two coordinate sets.

For a Riemannian metric on an ambient manifold and an embedded submanifold, let $\pi :TN{|}_M\to TM$ be orthogonal projection. Project the restriction of a connection to an embedded submanifold to obtain $D_{X}Y=\pi {\nabla }_{X}Y$. Since $\pi Y=Y$, the Leibniz rule and linearity over smooth functions hold, making $D$ an affine connection. If the ambient connection is the Levi-Civita connection, pairing with tangent vectors shows that $D$ is a metric connection; projecting the zero-torsion identity shows it is a torsion-free connection. By uniqueness, $D$ is the induced Levi-Civita connection.

For the Levi-Civita connection, the curvature form of a connection is the endomorphism-valued two-formIt has tensoriality in all three arguments. Its first Bianchi identity follows from torsion-freeness: the cyclic sum is ${\sum }_{\mathrm{cyc}}({\nabla }_X[Y,Z]-{\nabla }_{[Y,Z]}X)={\sum }_{\mathrm{cyc}}[X,[Y,Z]]=0$ by the Jacobi identity.

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 $f$.