I have four questions
-- It is mentioned that an n-dimensional space and an m-dimensional space may be used to determine a new and unique (n+m)-dimensional product space.
if we consider two circles of different radii, that are perpendicular to one another and intersect at a point, would produce a product space of a torus, that has 3-dimensions and not 4, dimensional space as per the statement.
What is wrong with the above case?
-- The component of a dyad, ${g}_{ij}$ in general is given by
${g}_{ij} = \vec{e}^i . \vec{e}^j$
where
$\vec{e}^i$ and $\vec{e}^j$ -- Contravariant basis vectors
a similar statement can be defined for a covariant basis vector
Why has the superscript of the contravariant basis replaced by the subscript for the dyad component, ${g}_{ij}$.
the same happens for the Kronecker delta,
${\delta}_{j}^i = \vec{e}^j . \vec{e}_i$
why are the indices exchanged for the Kronecker delta and for dyad component? Are their any significances or is it done as a part of mathematical symbols?
-- If the basis under consideration is not orthonormal or even orthogonal, then can it be said that the covariant basis and the contravariant basis are reciprocal..
The axes parallel to the local axes and perpendicular to the product surface will have an angle between them. Thus their dot product will never be the square of their magnitude, as in case if the orthonormal basis and the dot products of the other indices, $e_{i} . e^{j}$, if $i \neq j$ will be zero, as the angle would be $\pi/2$, as per the above defenition.
Then how can the two sets of basis vectors be always reciprocal(The general case).
-- Are metric tensors the tensors that are symmetrical tensors? Is there anything else to it, that merits any significance?