In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of the monodromy operator acting on the cohomology of a degenerating family of complex algebraic varieties).
The argument uses the base change compatibility and regularity of the Gauss-Manin connection to reduce the problem to the following statement: if $M$ is a complex square matrix such that, for every field automorphism $\sigma$ of $\mathbb{C}$, $\exp(2\pi i M^{\sigma})$ is conjugated to a matrix with integer coefficients, then $\exp(2\pi i M)$ is quasi-unipotent. This in turn is a simple consequence of Gelfond-Schneider theorem!
I always found this proof quite surprising and I was wondering if there aren't other unexpected applications of transcendental numbers out there.
Best Answer
Here is an application of transcendental number theory to differential geometry that I think would count as unexpected to all but a small group of experts in the area (who would probably view the application as being a very natural one).
Let $G$ be a connected absolutely simple real algebraic group and $\mathcal G=G(\mathbb R)$ the corresponding real Lie group. We'll call $\mathcal G$ absolutely simple. Let $\mathcal K$ be a maximal compact subgroup of $\mathcal G$ and $\mathfrak X=\mathcal K\backslash \mathcal G$ the symmetric space of $\mathcal G$.
If $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $\mathcal G$ then denote by $\mathfrak X_{\Gamma_1}=\mathfrak X/\Gamma_1$ and $\mathfrak X_{\Gamma_2}=\mathfrak X/\Gamma_2$ the associated locally symmetric spaces. We say that such a locally symmetric space is arithmetically defined if the corresponding discrete subgroup of $\mathcal G$ is arithmetic.
In their paper Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Gopal Prasad and Andrei Rapinchuk prove a number of interesting results about the spectral theory of such locally symmetric spaces, for instance:
Theorem (Prasad and Rapinchuk) Let $\mathfrak X_{\Gamma_1}$ and $\mathfrak X_{\Gamma_2}$ be two arithmetically defined locally symmetric spaces of the same absolutely simple real Lie group $\mathcal G$. If they are isospectral, then the compactness of one of them implies the compactness of the other.
Theorem (Prasad and Rapinchuk) Any two arithmetically defined compact isospectral locally symmetric spaces of an absolutely simple real Lie group of type other than $A_n (n > 1)$, $D_{2n+1} (n\geq 1)$, $D_4$ and $E_6$, are commensurable to each other.
While these results are unconditional for rank one locally symmetric spaces, for spaces of higher rank the results depend on Schanuel's conjecture in transcendental number theory.
As I mentioned above, I think that this geometric application would probably come as a complete surprise to non-experts, whereas to people working the field it is extremely natural, the idea being that the Laplace spectrum of such a space is related to the geodesic length spectrum (by results of Duistermaat and Guillemin), and the lengths of geodesics are in turn given by logarithms of algebraic numbers.