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.