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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?

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  • $\begingroup$ Re. your first question - a circle has dimension 1 (locally, it is equivalent to a straight line) and the product of two circles - a torus - has dimension 2. Remember we are only concerned with points on the surface of the torus. $\endgroup$ – gandalf61 Jan 10 at 15:51
  • $\begingroup$ Isn't the dimension related to the geometry.. like a torus being a 3D object in Cartesian and polar coordinates. A circle is R = constant in polar and is a 2D curve.. any point has to have 3 coordinates for a torus..So n = 2 and m= 2, so n+m = 4?? $\endgroup$ – Raptor Jan 10 at 15:55
  • $\begingroup$ You are confusing the dimension of an object itself with the dimension of the space in which it happens to be embedded. You only need one co-ordinate to specify a point on a circle (e.g. angle) therefore it has a dimension of 1. This is regardless of whether it is a circle embedded in 2D space or 3D space or even 4D space. If this is not clear then you should revise the concept of dimension. $\endgroup$ – gandalf61 Jan 10 at 16:08
  • $\begingroup$ True.. I an a Noob, looking for a book.. $\endgroup$ – Raptor Jan 10 at 16:09

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