Burgers Equation as a Logarithmic Connection: Gauge Structure, the Cole-Hopf Transformation and Prequantum Geometry

We present a geometric interpretation of the viscous Burgers equation in terms of an abelian logarithmic connection on a complex line bundle. The velocity field is shown to define the local component of a gauge potential, while the Cole-Hopf transformation corresponds to a logarithmic trivialization of the associated flat connection. In this framework, the nonlinear Burgers dynamics is reinterpreted as an evolution on the space of connections, whereas the Cole-Hopf transform induces a linear Schr？dinger-type (heat) equation on sections of the bundle. We analyze the resulting gauge structure and show that the construction can be organized using concepts from prequantum geometry, without implying a genuine quantization scheme. The flatness of the connection and the absence of a non-degenerate symplectic form are discussed as intrinsic obstructions to full quantization. This viewpoint clarifies both the linearization mechanism of the Cole-Hopf transformation and its geometric limitations. In this paper, no new analytical results are claimed.