Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

Question

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f :H^m \to H^n $ w.r.t the Hyperbolic metric in section 1, "Estimates on Douady-Earle extension" by using the connection forms as follows: (for me, I am happy with $m=n=2$, so I am just re-writing everything in $2$-dimensions). Let $ \theta_i, \theta_{ij} $ respectively denote the local orthonormal coframe on $H^2$ (or $D^2$) and its corresponding matrix of connection forms, $1\le i,j,\le 2$. For a map $\Phi: D^2 \to D^2$, one defines the energy density of $\Phi$ by $ e(\Phi)= \sum_{i,\alpha} {(f_{i}^{\alpha}})^2$, where $f_{i}^{\alpha}$'s are given by $F^*(\theta_\alpha)= \sum_{i}f_{i}^{\alpha}\theta_i $. Then according to this paper, the Hessian of $\Phi$ is denoted by $f_{ij}^{\alpha}$, is defined by :

$\sum_{j} f_{ij}^{\alpha}. \theta_j = df_i^{\alpha} - \sum_j{f_j^\alpha}.\theta_{ij} + \sum_{\beta}f_{i}^{\beta}F^*(\omega_ {\beta} {\alpha} ) $

and they write the norm of the Hessian w.r.t. this orthonormal coframe as :$|Hess(f)|^2=\sum_{ij\alpha} (f_{ij}^{\alpha})^2 $.

I guess I am a little confused by this definition. So my questions are :

1) Previously I knew ( from Riemmanian Geometry textbook of Peter Petersen or John Lee P.54 ) that $Hess(\Phi)=\nabla^2(\Phi) $ is a $(2,0)$ tensor defined by $Hess(\Phi)(X,Y)= g( \nabla_X(\nabla\Phi),Y) $, where $g$ denotes the hyperbolic metric on the open unit disk $D^2$.

Is the new definition of the paper the same as the old definition ? And what should be the norm according to the old definition ?

2) The proposition 1.2, (iii) states that under certain condition, the norm of the Hessian ( as defined before ) of $\Phi$ is $ < \epsilon $ .Does it mean that we can say the standard partial derivatives of $\Phi$, i.e. $\Phi_{xx}, \Phi{xy}, \Phi{yy} $ have norms less than $ < 4\epsilon ( 1- |z^2| )^2$ at the point $z$ under the same condition ( that condition is not necessary for my question here ) , given that we are working with $D$ with the standard Poincare metric on it ,i.e., $g= 4\frac{|dz|^2}{( 1- |z^2| )^2}$?

Thanks so much !

Answer 1

This just fleshes out Deane's comment.

Let $f: M \to N$ be a smooth map between two riemannian manifolds $(M,h)$ and $(N,g)$.

You can view $df$ as a one-form on $M$ with values in the pullback bundle $f^{-1}TN$. In other words, $df \in \Omega^1(M; f^{-1}TN)$ or if you prefer a smooth section of the vector bundle: $T^*M \otimes f^{-1}TN$ over $M$.

Both of these bundles have natural connections: on $T^*M$ you have the Levi-Civita connection associated to the metric $h$ on $M$, whereas on $f^{-1}TN$ you have the pullback of the Levi-Civita connection on $TN$ associated to the metric $g$ on $N$. The Hessian of $f$ is then simply $\nabla df$ where $\nabla$ is the product connection on $T^*M \otimes f^{-1}TN$.

This agrees, of course, with the equation you quote from the paper.