Part 1
Chapter 2 The Inverse Metric
For every Riemannian metric $g$, which is represented in local coordinates by the matrix of components $[g_{ij}]$, there exists an inverse metric. The components of the inverse metric are denoted with upper indices, $g^{ij}$, and are defined as the entries of the matrix inverse of $[g_{ij}]$.
\[[g^{ij}] = [g_{ij}]^{-1}\]As a tensor \(g^{-1} = g^{ij} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}\) is a well-defined symmetric $(2,0)$-tensor. As in Chapter 1, we use the Einstein summation convention.
Key Property
The fundamental relationship between the metric and its inverse is captured by the following equation, where $\delta_i^j$ is the Kronecker delta (the components of the identity matrix):
\[g_{ik}g^{kj} = \delta_i^j\]This is the formal statement that the two matrices are inverses of each other.
Application: Raising Indices
The primary algebraic use of the inverse metric is to “raise indices,” which provides a canonical way to convert a covector (a $(0,1)$-tensor) into a vector (a $(1,0)$-tensor).
If you have a covector with components $V_j$, you can obtain the components of the corresponding vector $V^i$ using the inverse metric:
\[V^i = g^{ij}V_j\]This operation is fundamental for defining many concepts in geometric analysis, such as the gradient of a function.