The drone navigation requires the comprehensive understanding of both visual and geometric information in the 3D world. In this paper, we present a Visual-Geometric Fusion Network (VGF-Net), a deep network for the fusion analysis of visual/geometric data and the construction of 2.5D height maps for simultaneous drone navigation in novel environments. Given an initial rough height map and a sequence of RGB images, our VGF-Net extracts the visual information of the scene, along with a sparse set of 3D keypoints that capture the geometric relationship between objects in the scene. Driven by the data, VGF-Net adaptively fuses visual and geometric information, forming a unified Visual-Geometric Representation. This representation is fed to a new Directional Attention Model (DAM), which helps enhance the visual-geometric object relationship and propagates the informative data to dynamically refine the height map and the corresponding keypoints. An entire end-to-end information fusion and mapping system is formed, demonstrating remarkable robustness and high accuracy on the autonomous drone navigation across complex indoor and large-scale outdoor scenes.