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For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.

For the Levi-Civita connection, the cyclic covariant derivative of curvature vanishes: ${\nabla }_{[e}{R}^a{}_{|b|cd]}=0$. Contracting this identity gives the contracted Bianchi identity.

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 $g=N\cdot X_{vv}$.

For $E=\frac{1}{2}\int |\dot{\gamma }{|}^2$ and a geodesic base curve, a variation field $V$ satisfiesThe endpoint term vanishes for fixed endpoints and for periodic variations of closed geodesics. The integral is the Riemannian index form. This uses the Levi-Civita connection and the convention $R(X,Y)={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$.

The spatial covariant derivative projects the spacetime covariant derivative and every free tensor index onto a spatial hypersurface. It is the Levi-Civita connection of the induced spatial metric.

The second fundamental form of an embedded submanifold of a Riemannian manifold is symmetric. Its ambient Levi-Civita connection is a torsion-free connection, soThe Lie bracket of vector fields tangent to the submanifold is tangent and therefore has zero normal component. This proof applies to a curved ambient manifold as well as Euclidean space.

For the ambient Levi-Civita connection, a normal field $\eta$, and tangent fields $X,Y$, differentiate $⟨\eta ,Y⟩=0$ using metric compatibility. The tangential part of ${\nabla }_{X}Y$ is orthogonal to $\eta$, giving the displayed identity. It controls how an ambient derivative of a normal vector can develop a tangential component, and is the vector-valued version of the shape operator relation.

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 $D_i\sqrt{h}=0$.