I am reading the first chapter of Information Geometry and its applications by Amari. I am struggling to grasp a basic concept about mutually orthogonal coordinate systems. Since the book is not available for free, I will elaborate on the problem in detail.
In the first chapter, the author introduces the definition of Manifold $M$ and one of the many coordinate systems on the manifold, say, $\theta$.
The length of the curve from $\boldsymbol {\theta}$ to $\boldsymbol{ \theta + d\theta}$ is given by $$ d s^{2}=2 D_{\psi}[\theta : \theta+d \theta]=\sum g_{i j} d \theta^{i} d \theta^{j} $$
A tangent vector can be expressed as $$ d \boldsymbol{\theta}=\sum d \theta^{i} \boldsymbol{e}_{i} $$ where $ \left\{\boldsymbol{e}_{i}\right\} $, $i \in \{1,.. ,n\}$ are the basis of the tangent space of M at point $\boldsymbol \theta$. Similarly, the author introduces a dual affine coordinate system whose corresponding basis is $\left\{e^{* i}\right\}$. Therefore,we can write
$$ d \boldsymbol \theta^{*}=\sum d \theta_{i}^{*} e^{* i} $$
Now, one can also write the length of the small line vector as $$ d s^{2}=\langle d \boldsymbol{\theta}, d \boldsymbol{\theta}\rangle= g_{i j} d \theta^{i} d \theta^{j} $$, which is rewritten as $$ d s^{2}=\left\langle d \theta^{i} e_{i}, d \theta^{j} e_{j}\right\rangle=\left\langle e_{i}, e_{j}\right\rangle d \theta^{i} d \theta^{j} $$ Hence, it is clear that $$ g_{i j}(\boldsymbol{\theta})=\left\langle\boldsymbol{e}_{i}, \boldsymbol{e}_{j}\right\rangle $$
Similarly, for the dual affine coordinate system $\boldsymbol \theta^*$, we have $$ g^{* i j}(\boldsymbol \theta^*)=\left\langle e^{* i}, e^{* j}\right\rangle $$
If $\bf G$ is the Jacobian of the transformation from $\boldsymbol \theta$ to $\boldsymbol \theta^*$, then we can write $$ \begin{array}{l}{d \boldsymbol{\theta}^{*}=\mathbf{G} d \boldsymbol{\theta}, \quad d \boldsymbol{\theta}=\mathbf{G}^{-1} d \boldsymbol{\theta}^{*}} \\ {d \theta_{i}^{*}=g_{i j} d \theta^{j}, \quad d \theta^{j}=g^{* j i} d \theta_{i}^{*}}\end{array} $$
I was able to follow till this point. I am unable to understanding how the author says the following:
The two bases $\left\{e_{ i}\right\}$ and $\left\{e^{* i}\right\}$ are mutually reciprocal or dual - because $$ e^{* i}=g^{i j} e_{j}, \quad e_{i}=g_{i j} e^{* j} $$ and hence, $$ \langle e_{i}, e^{* j}\rangle=\delta_{i}^{j} $$
What is the meaning of $\delta_{i}^{j}$? Can you please give a proof of how this follows?
Thank you,
This follows straight from the transformations that have been developed, namely
$$\left< \hat{e}_i, \hat{e}^{*j}\right> = (\sum_k g_{ik}\hat{e}^{*k}) \cdot \hat{e}^{*j} = \sum_k g_{ik}g^{kj} = \delta_i^j = \begin{cases} 0 \ i\neq j\\ 1 \ i = j \end{cases}$$
As the author states in the text, that the basis $\{ e_i \}$ or $\{ e^{i*} \}$ may not be individually orthogonal but they are mutually orthogonal.