A major difficulty in studying the neural mechanisms underlying olfactory perception is the lack of obvious structure in the relationship between odorants and the neural activity patterns they elicit. Here we use odor-evoked responses in piriform cortex to identify a latent manifold specifying latent distance relationships between olfactory stimuli. Our approach is based on the Gaussian process latent variable model, and seeks to map odorants to points in a low-dimensional embedding space, where distances between points in the embedding space relate to the similarity of population responses they elicit. The model is specified by an explicit continuous mapping from a latent embedding space to the space of high-dimensional neural population firing rates via nonlinear tuning curves, each parametrized by a Gaussian process. Population responses are then generated by the addition of correlated, odor-dependent Gaussian noise. We fit this model to large-scale calcium fluorescence imaging measurements of population activity in layers 2 and 3 of mouse piriform cortex following the presentation of a diverse set of odorants. The model identifies a low-dimensional embedding of each odor, and a smooth tuning curve over the latent embedding space that accurately captures each neuron's response to different odorants. The model captures both signal and noise correlations across more than 500 neurons. We validate the model using a cross-validation analysis known as co-smoothing to show that the model can accurately predict the responses of a population of held-out neurons to test odorants.