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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 using the Orthogonal Procrustes problem in the Euclidean space. This study presents a geometric framework that leverages smooth Riemannian manifolds for corpus-specific orthogonal rotations and a corpus-independent scaling metric 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.