# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting
## 2 Materials and Methods (Theoretical Background)
### Differential Manifold
#### 2.1.1 Metric or Riemannian Manifold  
A metric or Riemannian manifold is one where a notion of distance between any two points of the manifold is also implemented. This can be done using the definition of a global function in \(\mathbb{M}\) (tensor \(\binom{\kappa}{\lambda}\)).  
**Definition 1**.: _A Riemannian metric tensor \(\mathbf{g}:\mathbb{M}\times M\to\mathbb{R}\) is a type \(\binom{\kappa}{2}\) tensor that satisfies the following properties [56]:_  
1. \(\mathbf{g}(X,Y)\geq 0\)
2. \(\mathbf{g}(X,Y)=0\iff X=Y\)
3. \(\mathbf{g}(X,Y)=\mathbf{g}(Y,X)\)
4. \(\mathbf{g}(X,Z)\leq\mathbf{g}(X,Y)+\mathbf{g}(Y,Z)\)  
\(\forall\ X,Y,Z\in T_{p}(\mathbb{M})\)_._  
The components of the metric tensor \(\mathbf{g}\) with respect to the coordinate system \(\{q^{i}\}\) are  
\[g_{ij}=\mathbf{g}\big{(}\partial_{i},\partial_{j}\big{)}\, \tag{1}\]  
where \(\partial_{i}:=\
icefrac{{\partial}}{{\partial q_{i}}}\). The metric is a symmetric tensor, i.e., \(g_{ij}=g_{ji}\).  
**Definition 2**.: _A metric or Riemannian space \((\mathbb{M},\mathbf{g})\) is a manifold equipped with a metric tensor \(\mathbf{g}\)._Covariant Derivative  
Now it is important to define the differentiation in a Riemannian manifold. Clearly, in order to calculate the change of a vector along a curve (different points of a manifold), it should be considered that at each point \(p_{j}\in\mathbb{M}\), \(T_{p_{j}}(\mathbb{M})\) changes, and that change is reflected at the base of \(T_{p_{j}}(\mathbb{M})\) at each point. This change, due to the curvature, must be considered when calculating the derivative of a vector (or, in general, of any tensor) along an intrinsically curved manifold. The correction factor is always proportional to a set of coefficients, called Christoffel symbols, which in the coordinate system \(\{q^{i}\}\) can be determined through the components of the metric tensor (1) as  
\[\Gamma^{l}_{mn}=\frac{1}{2}\sum_{k=1}^{d}g^{lk}(\partial_{m}g_{mk}+\partial_{n }g_{mk}-\partial_{k}g_{mn})\, \tag{2}\]  
where \(l,m,n=1,\cdots,d\), \(g^{lk}\) will be the components of the inverse matrix of the metric coefficients (1). The Christoffel symbols are symmetrical in their two covariant indices, i.e., \(\Gamma^{l}_{mn}=\Gamma^{l}_{nm}\). In this way, the derivative of a vector will be given as follows.  
**Theorem 1**.: _Let \(\kappa:I\subset\mathbb{R}\to\mathbb{M}\) be a curve in \((\mathbb{M},\mathbf{g})\) (parametrized by \(\lambda\)), and let \(\mathrm{Y}_{\kappa}(\mathbb{M})\) be the space of all the smooth vector fields induced by \(\kappa\). The covariant derivative along \(\kappa\) will be the function:_  
\[\frac{D}{D\lambda}:\mathrm{Y}_{\kappa}(\mathbb{M})\to\mathrm{Y}_{\kappa}( \mathbb{M})\, \tag{3}\]  
_which satisfies the properties of linearity, additivity and the Leibniz rule as well._  
This definition can be generalised trivially for the derivative of any tensor type \(\binom{n}{N}\). In the coordinate system \(\{q^{i}\}\), the derivative (3) of the components of the vector \(\vec{V}\) vector will be given by  
\[\frac{DV^{i}}{D\lambda}=\sum_{j=1}^{d}U^{j}\partial_{j}V^{i}+\sum_{j,k=1}^{d} \Gamma^{i}_{kj}V^{k}U^{j}\, \tag{4}\]  
where \(\vec{U}=\frac{d\vec{q}}{d\lambda}\) is the vector tangent to the curve \(\kappa\).