Question about the metric tensor

Question

From wikipedia:

The metric can be written in the form $g=g_{ij}dx^i \otimes dx^j$. The metric is thus a linear combination of tensor products of one form gradients of coordinates. If we denote the symmetric tensor product by juxtaposition (so that $dx^i dx^j = dx^i dx^j$) we can write the metric in the form $g=g_{ij}dx^i dx^j$.

Can someone explain the last part; that is, how we go from a tensor product to a regular product?

Answer 1

We can use $g_{ij}dx^idx^j$ and $g_{ij}dx^i\odot dx^j$ interchangeably, because it is just matter of notation.

There is a notion of symmetric tensor and symmetric product of two tensors. In particular given two (0,1) tensors $a_idx^i,b_jdx^j$ one have a symmetric product of those two $$a_idx^i\odot b_jdx^j=a_ib_j\frac{1}{2}(dx^i\otimes dx^j+dx^j\otimes dx^i)$$ Aditionally any symmetric (0,2) tensor can be written in a form $$h_{ij}dx^i\odot dx^j$$

Let us now consider metric $g_{ij}dx^i\otimes dx^j.$ Just form definition of metric we have that $g_{ij}=g_{ji}.$ As a result $$g_{ij}dx^i\odot dx^j=g_{ij}\frac{1}{2}(dx^i\otimes dx^j+dx^j\otimes dx^i)=\\\frac{1}{2}(g_{ij}dx^i\otimes dx^j+g_{ji}dx^j\otimes dx^i)=g_{ij}dx^i\otimes dx^j.$$ We can put signs $=$ for following reasons.

• Firs due to the definition of symmetric product
• Second due to the fact that $g_{ij}=g_{ji}$
• Third due to the fact in summation we have only dummy indeces so we can change $i$ with $j$

Hence, in the situation when we are given with metric, we can say that we have $g_{ij}dx^i\odot dx^j$ or $g_{ij}dx^i\otimes dx^j$ such that $g_{ij}=g_{ji}.$