Can someone help me understand the Euclidean metric?

Question

A Euclidean metric is defined as:

$g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$

Can someone explain the following:

1. why do we use $dx^i$ instead of $x^i$ which is a coordinate in $R^n$
2. Why oes tensor product $\times$ gets turned into multiplication?
3. This does not look like a tensor at all, but rather just sum of products...

Finally, a metric is just an inner product. This does not look anything like an inner product! An inner product on Euclidean space is defined as $\langle \cdot , \cdot\rangle$, how is that metric thingy related to my inner product?

Answer 1

1. Consider the properties of the $dx^i$ compared to the $x^i$. As linear functionals, the basis one-forms $dx^i$ are...well, linear on their arguments. Inner products are bilinear, but we are using two one-forms in each linearly independent term.
2. Notational convention. My opinion? It's very misleading. I would almost always keep the tensor product explicit there.
3. It's a sum of tensor products, and it defines a symmetric, positive definite bilinear form. Can you see that for any vectors $a,b$, the quantity $g(a,b)$ obeys all the properties of an inner product of $a$ and $b$?