# CDSTAEP: Cross-Domain Spatial-Temporal Association Learning for Abnormal Events Prediction
## 3 Preliminaries and Problem Definition
### Riemannian Geometry  
A manifold \(\mathcal{M}\) of dimension \(n\) is a generalization to higher dimensions of the notion of surface and is a space that locally looks like \(\mathbb{R}^{n}\). Each point \(x\in\mathcal{M}\) associates with a tangent space \(Tx\mathcal{M}\), which is a vector space of dimension \(n\), that can be understood as a first order approximation of \(\mathcal{M}\) around \(x\). A Riemannian metric \(g\) is given by an inner-product \(g_{x}(.,.):Tx\mathcal{M}\times Tx\mathcal{M}\rightarrow\mathbb{R}\) at each tangent space \(Tx\mathcal{M}\), \(g_{x}\) is a symmetric, positive definite second-order covariant tensor on the tangent space \(Tx\mathcal{M}\), \(g_{x}\) varying smoothly with \(x\). Specifying a smooth manifold \(\mathcal{M}\) whose Riemannian metric is g is called a Riemannian manifold, denoted as \((\mathcal{M},g)\), or simply denoted as \(\mathcal{M}\). Intuitively, a Riemannian manifold behaves like a vector space only in its infinitesimal neighborhood, allowing the generalization of standard notation such as angle, straight line, and distance, to a smooth manifold. Exponential and logarithmic transformations can realize the mutual mapping between the local manifold space at a certain point in Riemannian space and the tangent space (Euclidean space) of the point. The logarithmic transformation maps the point (manifold space) to tangent space; after the corresponding geometric transformation is carried out using the Euclidean operator, the tangent space is mapped back to the manifold vector space by the exponential transformation. Where exponential mapping can be defined as \(exp_{x}(v):Tx\mathcal{M}\rightarrow\mathcal{M}\), logarithmic mapping can be defined as \(log_{x}(y):\mathcal{M}\rightarrowTx\mathcal{M}\).  
**Riemannian Metric:** Riemann distance is a curve. The distance (Riemannian metric) in Riemannian space can be obtained in tangent space first, that is, first specify the length for the tangent vector on the manifold, and then define the length of the curve as the integral of the length of the tangent vector along the curve, and then mapped back to the manifold space by the exponential map \(exp_{x}(v):Tx\mathcal{M}\rightarrow\mathcal{M}\). In the Riemannian space \(\mathcal{M}\), the general expression of the square distance formula at \(X=(x_{1},\cdots,x_{n})\) is as follows:  
\[d_{X}^{2}(x,y)=\sum_{i}d_{i}^{2}(x_{i},y_{i}) \tag{1}\]  
**Product Manifolds:** For a smooth manifold sequence \(\mathcal{M}_{1},\mathcal{M}_{2},\cdots,\mathcal{M}_{n}\), the Riemann product manifold can be defined as a Cartesian product [9]\(\mathcal{M}=\mathcal{M}_{1}\times\mathcal{M}_{2}\times\cdots\times\mathcal{M}_ {n}\). For any point \(x\in\mathcal{M}\), it can be expressed symbolically as \(x=(x_{1},x_{2},\cdots,x_{n}):x_{i}\in\mathcal{M}_{i}\), such as \(\mathcal{M}_{3}=\mathcal{M}_{1}\times\mathcal{M}_{2}\), which means that \(\mathcal{M}_{3}\) is given by the point \(x_{1}\) on \(\mathcal{M}_{1}\) and the point \(x_{2}\) on \(\mathcal{M}_{2}\). Similarly, the tangent vector space \(v\in Tx\mathcal{M}\) of point \(x\) can be defined as \(v=(v_{1},v_{2},\cdots,v_{n}):v_{i}\in Tx_{i}\mathcal{M}_{i}\), that is, the tangent vector in the product manifold is given together by the tangent vectors of each sub-manifold.