Title: Design of the topology for contrastive visual-textual alignment. Abstract: Pre-training weakly related image-text pairs in the contrastive style shows great power in learning semantic aligning cross-modal models. The common choice to measure the distance between the feature representations of the image-text pairs is the cosine similarity, which can be considered as the negative inner product distance of features embedded on a sphere, mathematically. However, empirically, aligning image-text pairs on the spherical topology is vulnerable to the semantic ambiguity phenomenon resulting from the noise in the pre-training datasets. Specifically, under the noisy training data, instead of the optimal alignment-uniformity solution, the system would achieve an equilibrium (a gap between distances of positive and negative pairs), when the gradients for attraction and repulsion are neutralized.  Although intuitively, the model should always find this equilibrium given a sufficiently long training scheme, its numerical values might be out of the distance range (e.g. [-1, 1] for the cosine similarity). In the practice of former studies, this problem is partly tackled by introducing a learnable softmax temperature parameter, in other words, by explicitly scaling the range of the distance function.  In this work, we alternatively design the topology of embedding space and its endowed distance function. Motivated by studies that make use of Riemannian geometry for visual tasks, we propose a rather simple solution to address the aforementioned equilibrium problem. That is, we map the feature representations onto the oblique manifold endowed with the negative inner product as the distance function. In the experimental analysis, we show that we can improve the baseline performance by a large margin (e.g. 4\% in the zero-shot image to text retrieval task) by changing only two lines of the training codes.