We consider the problem of representing compact geometric data structures maintaining an efficient implementation of navigation operations. For the case of planar triangulations with m faces, we propose a compact representation of the connectivity information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports navigation between adjacent triangles in constant time. For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g-1)\lgm bits. The structure also allows constant time access to vertex specific data, like coordinates, but the paper does not address the compression of this geometric information
