# Question about the physical intuition behind tensors

I would like to check something with you. I am a beginner in differential geometry and in general relativity so I may say wrong things.

This is what I understood from tensors.

Tensors are object for which coordinates follows a specific transformation when a change of basis is done.

If I define a tensor on a basis $A$, I automatically know its components on a basis $B$.

The philosophy behind the object is to be able to define quantities that are "absolute" and don't depend on coordinates.

For example $ds^2=dx^2+dy^2$ is also written in polar coordinates $ds^2=dr^2+r^2 d\theta^2$ but it represents physically the same distance : if I compute the scalar product between two vectors using either the first or the second basis, the result will be the same.

So finally : can I say that tensors are mathematical object that are used to describe phenomenon that are absolute and don't depend on a given set of basis. For example if I take a n-contravariant and p-covariant tensor and I apply it on n vectors and p covectors the final result will not depend of any choice of basis.

Also : why do we only define tensors as object that transforms as $m'^i_j=m^k_l \frac{\partial x^l}{\partial y^j}\frac{\partial y^i}{\partial x^k}$ and not more general transformations ?

Is it because the tensors are constructed around the notion of vectors/covectors and we want that tensors applied on vector and covector give result independant of the coordinates. So all the tensors were constructed to ensure that applied on vector and covectors they give result indepent of coordinates.

Indeed, we could imagine to create other objects that ensure basis-independant results with other object than vectors but it is just that we don't need them in practice (in general relativity we would'nt need them for example).

My last question is juste to be sure that I understand well... or not.

Edit: to use a more physical language, tensors are fields on spacetime which vary in some sense in a way compatible to how the spacetime curves, much like, say, a tangent vector field varies on spacetime, or, say, a 1-form field varies on spacetime. This is the intuitive meaning of the transformation laws, and explains why tensors are so relevant to the geometry of spacetime. Examples of tensors include the Riemann curvature tensor, for instance, and the Ricci tensor, which is a contraction of the Riemann curvature tensor. The metric $g$ itself is a tensor. A connection is not a tensor, but the difference of two connections on a manifold is a tensor.