Comparing two bodies of text and detecting words with significant lexical semantic shift between them is an important part of digital humanities. Traditional approaches have relied on aligning the different embeddings in the Euclidean space using the Orthogonal Procrustes problem. This study presents a geometric framework that leverages optimization on smooth Riemannian manifolds for obtaining corpus-specific orthogonal rotations and a corpus-independent scaling to project the different vector spaces into a shared latent space. This enables us to capture any affine relationship between the embedding spaces while utilising the rich geometry of smooth manifolds.