Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians

Let $M$ be a smooth manifold.

• A Riemannian metric on $M$ is a smooth bundle metric on the tangent bundle $TM$ (Smooth bundle metrics).
• A cotangent-bundle connection is an $\mathbb{R}$-bilinear assignment $(X,\alpha )\mapsto {\mathrm{\nabla }}_X\alpha$ from smooth vector fields $X$ and smooth one-forms $\alpha$ to smooth one-forms such that ${\mathrm{\nabla }}_{hX}\alpha =h{\mathrm{\nabla }}_X\alpha ,{\mathrm{\nabla }}_X(h\alpha )=X(h)\alpha +h{\mathrm{\nabla }}_X\alpha$ for every smooth function $h$.

In a smooth chart $x=(x^1,\dots ,x^n)$, such a connection is determined by its Christoffel symbols ${\mathrm{\Gamma }}_{ij}^k$ through

${\mathrm{\nabla }}_{{\mathrm{\partial }}_{x^i}}(d{x}^k)=-{\sum }_j{\mathrm{\Gamma }}_{ij}^{k}d{x}^j,$

or equivalently, for $\alpha ={\sum }_j{\alpha }_{j}d{x}^j$,

${\mathrm{\nabla }}_{{\mathrm{\partial }}_{x^i}}\alpha ={\sum }_j(\frac{\mathrm{\partial }{\alpha }_j}{\mathrm{\partial }{x}^i}-{\sum }_k{\mathrm{\Gamma }}_{ij}^k{\alpha }_k)d{x}^j\mathrm{.}$

If $g$ is a Riemannian metric with local coefficients $g_{ij}:=g({\mathrm{\partial }}_{x^i},{\mathrm{\partial }}_{x^j})$, write ${\mathrm{\Gamma }}_{ijk}:={\sum }_{\mathrm{\ell }}{g}_{k\mathrm{\ell }}{\mathrm{\Gamma }}_{ij}^{\mathrm{\ell }}$. The connection is

• symmetric when ${\mathrm{\Gamma }}_{ij}^k={\mathrm{\Gamma }}_{ji}^k$ in every chart;
• metric-compatible with $g$ when $\frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^i}={\mathrm{\Gamma }}_{ijk}+{\mathrm{\Gamma }}_{ikj}$ in every chart.

For a smooth function $f:M\to \mathbb{R}$, the tensor

${\mathrm{\nabla }}^{2}f:=\mathrm{\nabla }(df)$

is the covariant Hessian of $f$. In coordinates, if $g=f\circ x^{-1}$,

$({\mathrm{\nabla }}^{2}f{)}_{ij}=\frac{{\mathrm{\partial }}^{2}g}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}-{\sum }_k{\mathrm{\Gamma }}_{ij}^k\frac{\mathrm{\partial }g}{\mathrm{\partial }{x}^k}\mathrm{.}$