Covariant derivative

A connection defines covariant differentiation by adding one connection term for each contravariant index and subtracting one for each covariant index.

The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.

The covariant divergence contracts a covariant-derivative index with a contravariant tensor index. For a mixed rank-two tensor it is ${\nabla }_{\mu }{T}^{\mu }{}_{\nu }$.

A tensor of type $(r,s)$ has $r$ contravariant and $s$ covariant indices. Taking a covariant derivative adds one covariant index, while an outer tensor product adds the corresponding contravariant and covariant index counts.

Under a coordinate change $x\mapsto x^{\prime }$, the components of a type-$(r,s)$ tensor acquire one forward Jacobian for each contravariant index and one inverse Jacobian for each covariant index. For a type-$(1,1)$ tensor,

A tensor density of weight $w$ transforms like a tensor with an additional factor $|\det (\partial x/\partial x^{\prime }){|}^w$ under $x\mapsto x^{\prime }$. A scalar density of weight one can be integrated against the coordinate volume element to give an invariant integral. On a spatial hypersurface, $\sqrt{h}$ is such a scalar density; hence ${\pi }^{ij}=\sqrt{h}{p}^{ij}$ is a tensor density when $p^{ij}$ is an ordinary tensor. The Levi-Civita connection extends to densities and satisfies $D_i\sqrt{h}=0$.

The identity tensor is the canonical type-$(1,1)$ tensor that maps each tangent vector to itself. It pairs a covector $\eta$ and vector $V$ as $\eta (V)$, and its components are the Kronecker delta ${\delta }^{\mu }{}_{\nu }$ in every coordinate system.

For $W_{\mu \nu }$,

The Levi-Civita connection is the unique torsion-free connection compatible with the metric; its coefficients are the Christoffel symbols formed from first metric derivatives.

Two Riemannian metrics have the same Levi-Civita connection precisely when their relative positive self-adjoint endomorphism $A$ is parallel for that connection. Parallel transport preserves both metrics. Consequently agreement at one point implies agreement everywhere on a connected manifold. Without that agreement, constant scalar multiples always give examples; constant nonproportional Euclidean forms show that a common connection does not force scalar proportionality.

A Riemannian metric determines a unique torsion-free connection that is a metric connection. Metric compatibility and the torsion identity imply the Koszul formula, which proves uniqueness. Conversely the right-hand side of the Koszul formula is a smooth one-form in its last vector-field argument. The musical isomorphism defines $D_{X}Y$ from it; expanding the Lie bracket of vector fields verifies the connection rules, metric compatibility and zero torsion. Thus it constructs the Levi-Civita connection intrinsically.

Metric compatibility is ${\nabla }_{\alpha }{g}_{\mu \nu }=0$, allowing the metric to pass through covariant derivatives and index operations.

The Lie derivative of a connection is the tensorAn isometry preserves the Levi-Civita connection, so ${\mathcal{L}}_K\nabla =0$ for a Killing vector field.