general relativity - Are fully raised/lowered versions of Kronecker delta tensors?

I am confused. I have two textbooks contradicting each other, at least, it seems to me so. The first one – "Field theory" by Landau & Lifshitz says that by lowering or raising one index of Kronecker delta one gets covariant/contravariant metric tensor(in Minkowski space). The second one "Introducing Einstein’s Relativity (1992)" by Ray d’Inverno shows the opposite, namely, by lowering or raising an index of Kronecker delta one gets an object which apparently is not a tensor( the last sentence says that in this link). Is there a contradiction or the problem is with my understanding?

Answer

The metric tensor $g : TM\times TM \to \mathbb{R}$ is by definition a $(0,2)$-tensor and transforms like one.

You question is not about the metric tensor, but about the Kronecker delta, and for which index positions it defines a tensor.

To define a tensor by its components, we must fix a coordinate system $x$ on our manifold $M$ since the components of a $(0,2)$-tensor $T$, for instance, are defined as coefficients in the expansion of $T$ in the basic tensors $\mathrm{d}x^\mu$:

$$T= T_{\mu\nu}\mathrm{d}x^\mu\otimes\mathrm{d}x^\nu$$ and we now want to examine for which index positions on the $\delta$ the tensors $$\delta^{ab}\partial_a\otimes\partial_b\quad\text{and}\quad\delta_{ab}\mathrm{d}x^a\otimes\mathrm{d}x^b\quad\text{and}\quad{\delta^a}_b\mathrm{d}x^a\otimes\partial_b$$ are defined independent of the chosen coordinate system.

To that end, we examine the transformation behaviour of $\delta^{ab},{\delta^a}_b,\delta_{ab}$. What we find is that under a coordinate transformation $x\mapsto y(x)$,

$$\begin{align} \delta^{ab} & \mapsto \sum_i \frac{\partial y^a}{\partial x^i}\frac{\partial y^b}{\partial x^i}\\ \delta_{ab} & \mapsto \sum_i \frac{\partial x^i}{\partial y^a}\frac{\partial x^i}{\partial y^b}\\ {\delta^a}_b &\mapsto {\delta^a}_b \end{align}$$

Therefore, a tensor that has components $\delta^{ab}$ or $\delta_{ab}$ in one coordinate system does not have those components in another system, so just writing down $\delta^{ab}$ does not specify a tensor unless you also specify a coordinate system in which the tensor has these components.

On the other hand, a $(1,1)$-tensor that has components ${\delta^a}_b$ in one system has those in all, therefore, ${\delta^a}_b$ defines a tensor without need for a particular coordinate system.

Now, since it defines a $(1,1)$-tensor, we can raise and lower its indices. However, contrary to what one might expect, the fully raised and fully lowered versions are not $\delta^{ab}$ and $\delta_{ab}$, but instead $g^{ab}$ and $g_{ab}$.

So, lowering/raising the index on the $\delta$ with the metric tensor¹ gives proper tensors, "raising/lowering" the index by just writing both indices rasied on the $\delta$ does not give a well-defined tensor.

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¹Here, "raising/lowering" with the metric tensor refers to the musical isomorphism that a vector $v = v^\mu\partial_\mu$ has an associated covector $v^\flat = g_{\mu\nu}v^\nu\mathrm{d}x^\mu$ with components $(v^\flat)_\mu = g_{\mu\nu}v^\nu$, and similarily lowering an index of a tensor means contracting it with $g_{\mu\nu}$, and raising it contracting it with $g^{\mu\nu}$. So, lowering the index of ${\delta^a}_b$ means $(\delta^\flat)_{ab} = g_{ac}{\delta^c}_b = g_{ab}$ by definition of the Kronecker delta.