A classical (linear) code is a linear subspace $\mathbb{Z}_2^k \subset \mathbb{Z}_2^n$, it encodes $k$ logical bits on $n$ physical bits. Similarly, a quantum error correction code encoding $k$ logical qubits on $n$ physical qubits is a complex linear subspace $L$ of dimension $2^k$ in the  space $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \dots \otimes \mathbb{C}^2 \cong \mathbb{C}^{2^n}$ of $n$-qubit states. In the particular case of stabilizer codes, $L$ is defined as the common 1-eigenspace of a suitable subgroup of unitary matrices, called stabilizer group. Most famous examples are the topological codes, like toric codes, where the qubits are associated to the edges of a cellulation of a 2-manifold, and the operators acting on the code space $L$ correspond to the elements of the first homology. Briefly speaking, the code distance is the minimal number of qubits which should be modified in a code word to get another code word. We reformulate the code distance in terms of differential geometry. The sum of a set of generators of the stabilizer group defines a hermitian matrix called code Hamiltonian with a $2^k$-fold eigenvalue degeneracy corresponding to the code space $L$. We will see that the code distance is essentially equal to the minimal order of distancing from the manifold of $2^k$-fold degenerate matrices at special directions corresponding to the operators which act non-trivially on only one qubit. This relation opens a new approach to study stabilizer codes via the `stickiness structure' of the degeneracy submanifold of matrices. Joint work with Andr\'{a}s P\'{a}lyi and D\'{a}niel Varjas.