A First Course in String Theory/Invariant Interval/Metric

[Cosmology2015]

Hello,
Before starting, I would like to apologize for any errors in the use of symbols. This is my first time .
I am studying the wonderful book of Barton Zwiebach, "A First Course in StringTheory".
In chapter 02, I am experiencing for the first time with the mathematics of special relativity (Minkowski Spacetime).
My question is on the definition of invariant interval ds[2]. By definition, the invariant interval is given by -ds[2]=η[μν]dx[μ]dx[ν]
I am not able to understand the minus sign on ds[2]. Is there any relationship with the idea of positive-definite condition? Others books use only ds[2] for the invariant interval. Is there any advantage in using this convention?
Another question would be about the invariant interval -ds[2.]. The definition of the invariant interval is very similar to the definition of Riemannian metric (metric tensor) g[ij].
(a) invariant interval → -ds[2]=η[μν]dx[μ]dx[ν]
(b) Riemannian metric → g=∑g[ij]dx⊗dx[j]
Is there any direct relationship? What is the difference between them?
I sincerely thank any reply .

 

[Cosmology2015]

Hello,
I would like to apologize for the errors in the use of symbols. As I told before, it was my first time, and I am still learning how to use the resources of this forum.
I sincerely thanks any reply .

 

[Ruslan_Sharipov]

In your case you have not the Riemannian metric, but the Minkovski metric. In writing any metric as a sum \sum_{ij}\,g_{ij}\ dx^i\otimes dx^j the differentials dx^i and dx^j are formal symbols. The interval notation ds or ds^2 is used if you want to calculate the length of a parametric curve x^i=x^i(\theta), \theta\in [0,1], in the Minkovski space. In this case you write dx^i=(x^i)'_\theta\,d\theta and then integrate s=\int^1_0 ds.