The question seems to conflate many different things:

- the invariance of a mathematical quantity (usually a scalar such as $ds^2$ for the separation of two events in special relativity)
- covariance of tensors (the values of components of tensors may be calculated from those in another frame but they're not the same thing)
- universality of equations in different situations (the same equations – defining a theory – have many solutions and different solutions e.g. different shapes of manifolds are generally not related to each other at all)

These things are perhaps related and resemble each other but they're not the same things. In special relativity, some objects such as $p_\mu p^\mu$ for an energy-momentum vector $p^\mu$ are "invariant" which really means that the value of this scalar quantity doesn't change at all if one performs a Lorentz transformation $L$:
$$ L(p)^\mu L(p)_\mu = p^\mu p_\mu $$
Then there are tensors which are any objects that transform "covariantly":
$$ L(T)_{\alpha\beta\dots \omega} = T_{\alpha' \beta' \dots \omega'} L^{\alpha'}_{\alpha} L^{\beta'}_{\beta} \dots L^{\omega'}_{\omega} $$
which means that they transform as "tensor products of vectors": each index is being contracted with a copy of the Lorentz transformation's matrix.

Field equations in special relativity are covariant: they (after all terms are moved to the left hand side and the right hand side vanishes) transform as tensors which means that if they vanish (hold) in one reference frame, they also do in another. However, the particular numerical values of the components of a (covariant) tensor do depend on the reference frame. They're not "invariant" (unchanging); instead, they're just "covariant" (they change together with the coordinates, according to a universal tensor rule).

In general relativity, the fields such as the Ricci tensor are functions of the spacetime coordinates. At each point, the objects transform as tensors (as explained above) under coordinate transformations that reduce to Lorentz transformations in the vicinity of the given point (up to a certain approximation). In fact, the tensor transformation rule above may be generalized and should be generalized from $SO(3,1)$ to $GL(4,R)$. This is also useful to write down how the tensor fields transform under general diffeomorphisms i.e. not necessarily linear coordinate transformations. For general coordinate transformations, the definition of a "tensor" is more constraining: for example, partial derivatives of vectors no longer transform as tensors.

With this more constraining definition, general relativity dictates field equations that have the form "tensor field vanishes". For a given theory, the equations of motion have the universal form – e.g. the Maxwell-Einstein equations, to be specific. The well-definedness and the uniqueness of the equations of motion is what we mean by having a single theory. However, a single theory or a single set of equations in physics always has many solutions. In special relativity, one may produce new (mathematically but not physically new) solutions by Lorentz transformations from a given one; in general relativity, one may obtain new (mathematically but not physically new) solutions by any diffeomorphisms applied on a given solution.

The tensor fields transform covariantly but they are not invariant and they depend on the situation – on the shape of the manifold etc. At any rate, at this point, you should understand why your question makes no sense. The mathematical foundation of "such application" is elementary linear algebra, differential geometry, special relativity, or general relativity, depending on what you're exactly asking about. However, you're not exactly asking about anything so your question can't be answered. Be sure that there doesn't exist any contradiction along the lines that you apparently wanted to propose in the wording of your question.

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