  The Weyl geometry promises potential applications in gravity and quantum
mechanics. We study the relationships between the Weyl geometry, quantum
entropy and quantum entanglement based on the Weyl geometry endowing the
Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl
scalar curvature in the $n$-dimensional system. The Weyl scalar field plays a
bridge role to connect the Weyl scalar curvature, quantum potential and quantum
entanglement. We also give the Einstein-Weyl tensor and the generalized field
equation in 3D vacuum case, which reveals the relationship between Weyl
geometry and quantum potential. Particularly, we find that the correspondence
between the Weyl scalar curvature and quantum potential is dimension-dependent
and works only for the 3D space, which reveals a clue to quantize gravity and a
understanding why our space must be 3D if quantum gravity is compatible with
quantum mechanics. We analyze numerically a typical example of two orthogonal
oscillators to reveal the relationships between the Weyl scalar curvature,
quantum potential and quantum entanglement based on this formulation. We find
that the Weyl scalar curvature shows a negative dip peak for separate state but
becomes a positive peak for the entangled state near original point region,
which can be regarded as a geometric signal to detect quantum entanglement.

General Relativity and Quantum Cosmology