inequalities on Riemmanian manifolds in local coordiantes Given a compact Riemmanian manifold $(X, g)$ and a smooth function $f$ on it. Recall that $\nabla f = g^{i j} \partial_{i} f \partial_{j}$, so $|\nabla f|^2 = g_{jq}g^{ij}g^{pq}\partial_{i} f \partial_{p} f$.
We can compute in normal coordinates that $\Delta_{g} |\nabla f |^2 = \partial_{k} \partial_{k}((g_{jq}g^{ij}g^{pq})) \partial_{i} f \partial_{p} f +$ a bunch of other terms.  Let me denote this first term on the right hand side by $R$. I have the following computation to get a lower bound for $R$:
$R \geq -C \sum |\partial_{i} f| |\partial_{p} f| \geq - C (\sum_{i} |\partial_{i} f|^2 + \sum_{p} |\partial_{p}f|^2) =  -C |\nabla f|^2$. This $C$ depends on some curvature of the metric.
I am not so sure about this computation and specifically my question is:

*

*I found online that if the manifold is compact, then one can treat the norm of a tensor $T$ using the norm of their components in local coordinates. This is equivalent to the norm defined as $|\nabla^{k} T|_{g}$. However, if we use the norm in local coordinates, it is going to be dependent on the choice of coordinates. What does it mean for it to be equivalent to the second definition?


*How do you treat inequalities in local coordinates? In the example I gave, can I just bound any random terms with constants like I did for the derivatives of the metric or does it have to be a tensor (globally defined objects) for me to throw it away using its bound.
1 is related to 2 because 1 seems to be suggesting that invariance does not matter much but that does not make sense to me.
 A: The first thing to note is that $g_{jq}g^{pq} = \delta_j^p$, so
$$ |\nabla f|^2 = g^{ij}\partial_if\partial_jf $$
Second, if you want a lower bound for $\Delta|\nabla f|^2$, it follows directly from the Bochner formula.
As for your questions about estimates for the norm of a tensor, it depends on what the situation is. If you're working on a fixed compact Riemannian manifold, and you need an estimate for the norm of a tensor that's independent of the tensor itself, then using the local coordinate version is fine. If, however, you want an estimate where the constants do not depend on the specific manifold but only on geometric properties of the manifold, then it's possible to still use local coordinates (but carefully) but it's usually better to find coordinate-free estimates of the norm of the tensor with respect to the metric. The Bochner formula is an example of this approach.
A: 
I found online that if the manifold is compact, then one can treat the norm of a tensor  using the norm of their components in local coordinates. This is equivalent to the norm defined as |∇|. However, if we use the norm in local coordinates, it is going to be dependent on the choice of coordinates. What does it mean for it to be equivalent to the second definition?

Potentially what is meant here is that there exists constants $C_0, C_1>0$ such that
$$C_0 \sum_{|I|=k} T_I^2\leq |\nabla^kT|^2_g \leq C_1\sum_{|I|=k} T_I^2$$
where $I$ denotes a multi-index of length $|I|=k$. (Notice that $C_0$ and $C_1$ here are independent of $x\in M$). On the left and right hand side, we have the Frobenius norm computed using the coordinate components of the tensor.
I believe this result is not independent of coordinates. However, it is independent of choice of normal coordinates. This should not be too surpising, since normal coordinates carry some information about the Riemannian metric on $M$, while general coordinates are not guaranteed to do so (although I believe the result might also hold for harmonic coordinates as well.)
This can be shown in the case $k=1$ using the two equivalent definitions of manifolds of bounded geometry.
A manifold of $C^k$-bounded geometry is a manifold $(M,g)$ with positive injectivity radius on which for each $i\leq k$, the $k$-th covariant derivative of the Riemannian curvature $|\nabla^i R|_g$ is bounded by a constant. Thus, a compact manifold is of $C^k$-bounded geometry for any choice of $k$.
Suppose that $(M,g)$ is of $C^k$-bounded geometry, then for $i\leq k$, there exists a $C_0, C_1>0$ such that in any normal coordinate chart of size less than injectivity radius,
$$
C_0 \leq |D^\alpha g_{ij}| \leq C_1 \text{ and }C_0 \leq |D^\alpha g^{ij}|\leq C_1
$$
where $D^\alpha$ indicates the ordinary derivative with multi-index $\alpha$ such that $|\alpha|\leq k$ in $\mathbb{R}^d$ taken in coordinates.
A classical result (I believe of Eichorn, but I need to track down the original paper) shows that the above property is actually equivalent to the bounded curvature condition above. Thus, if $(M,g)$ is a compact manifold, one has bounds on the components $g_{ij}$ and $g^{ij}$ as well as their derivatives in any normal coordinate chart.
Applying this to your original question, observe the norm of the gradient in a normal coordinate chart:
$$
|\nabla f|_g= g_{pq}g^{iq}g^{jp}\partial_i f \partial_j.
$$
Observe that for each $x$ in the domain of the coordinate chart, $g_{ij}(x)$ is a symmetric positive definite matrix. Since we have bounds on the entrees of these matrices which are independent of $x$, we have an upper bound on the maximum eigenvalue of $g_{ij}$ and a lower positive bound on the minimum eigenvalue by taking the maximum eigenvalue of $g^{ij}$. In other words, there is a $K_0,K_1>0$ such that for any vector $s= (s^1,...,s^d)$
$$
K_0 \delta_{ij}s^is^j \leq g_{ij}(x) s^is^j\leq K_1 \delta_{ij}s^is^j.$$
Using this together with the definition of the norm you gave, we get some $K'_0,K'_1>0$ only dependent on the bounded geometry constants and the dimension of $M$ such that
$$K'_0 \sum_i (\partial_i f(x))^2\leq |\nabla f(x)| \leq K'_1 \sum_i (\partial_i f(x))^2.$$
Where the derivatives on the left and rightmost sides are taken in any choice of normal coordinate chart on $M$ and the constants are independent of this choice. This is the Frobenius norm of the coordinate Jacobian.
A note that for higher-order covariant derivatives this becomes even more messy, since the covariant derivative of order $k$ in coordinates contains derivatives of  $f$ of all orders from $1$ to $k$. This means that a very messy inductive argument must be used. I am not sure if this has been written down, but I intend to write up some notes in the future on this.
