<p>The higher-dimensional <i>b</i>-equation is a family of PDEs, introduced by Holm and Staley (SIAM J Appl Dyn Syst 2(3):323–380, 2003), that describe the motion of shallow water waves in <i>n</i>-dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid, where the constant <i>b</i> can be thought of as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and show that this connection is Levi-Civita with respect to a right-invariant Riemannian metric only when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(b=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Using this framework and the methods of Ebin and Marsden (Ann Math (2):92:102–163, 1970), we show local well-posedness of the <i>b</i>-equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the <i>b</i>-equation as a smooth ODE on a Hilbert manifold, applying Picard–Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution.</p>