Smooth covariant 2-tensor fields on a smooth manifold

Let $M$ be a smooth $n$-manifold (Smooth manifolds and their smooth charts), with smooth atlas $\mathcal{A}$ and maximal smooth atlas $[\mathcal{A}]$ (Smooth atlases, Smoothly compatible charts and the smoothness of Euclidean transition maps). For $p\in M$ the tangent space $T_{p}M$ is the real vector space of derivations at $p$ (Derivations at a point and the tangent space), and the cotangent space is its algebraic dual $T_p^{\ast }M:=(T_{p}M{)}^{\ast }$ (Cotangent space and cotangent bundle as a disjoint union).

A covariant $2$-tensor at $p$ is a bilinear map $A_p:T_{p}M\times T_{p}M\to \mathbb{R}$. It is symmetric when $A_p(v,w)=A_p(w,v)$ for all $v,w\in T_{p}M$, and nondegenerate when the only $v\in T_{p}M$ with $A_p(v,w)=0$ for all $w\in T_{p}M$ is $v=0$.

A smooth covariant $2$-tensor field on $M$ is a family $A=(A_p{)}_{p\in M}$ assigning to each $p$ a covariant $2$-tensor $A_p$ at $p$, such that for every chart $(U,x)$ of the maximal smooth atlas $[\mathcal{A}]$ the component functions

$A_{ab}(q):=A_q\text{\hspace{0.17em}⁣}(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}{\mid }_q,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^b}{\mid }_q),q\in U,$

are $C^{\mathrm{\infty }}$ on $U$. Here $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a{\mathrm{\mid }}_q$ is the coordinate derivation at $q$ (Derivations at a point and the tangent space, Coordinate derivations form a basis of the tangent space), the index ranges over the label set $I_n=\{1,\dots ,n\}$ (or the relabelled set $I_4=\{0,1,2,3\}$ when $n=4$), and smoothness of the real-valued functions $A_{ab}$ on the open set $U\subseteq M$ is the notion of $C^r$ and smooth maps between smooth manifolds applied to $M$ and $\mathbb{R}$.

For a coordinate tuple $z\in x(U)$ define the coordinate representative $A_{ab}^x(z):=A_{ab}(x^{-1}(z))$. When the argument is a coordinate tuple, $A_{ab}(z)$ abbreviates this representative in the named chart; when it is a point of $M$, it denotes the component function on the chart domain. Thus $A_{ab}(x(y^{-1}(u)))$ in a change-of-coordinates formula means $A_{ab}^x((x\circ y^{-1})(u))$, never evaluation of a function on $M$ at a tuple. For $n=4$, $x^a$ and ${\mathrm{\partial }}_a$ mean the original coordinate and derivation labelled $a+1$, respectively; this relabels every slot and sum. The label $0$ alone gives no coordinate vector a causal character.

Because the coordinate derivations at $q$ form a basis of $T_{q}M$ (Coordinate derivations form a basis of the tangent space), specifying $A$ is the same as specifying the matrix of component functions in each chart; Chart independence and the transformation law of a smooth covariant 2-tensor field ↗ proves that the smoothness requirement is independent of the chart used, so that the condition may be checked on any smooth atlas of $M$, and derives the transformation law relating components in two overlapping charts.

The field is symmetric when $A_p$ is symmetric for every $p\in M$, which in a chart is the identity $A_{ab}=A_{ba}$ of component functions; it is nondegenerate when $A_p$ is nondegenerate for every $p$, which in a chart is the nonvanishing at every point of the polynomial function $\det (A_{ab})$ of the components. The matrix of $A$ in a chart means the matrix $(A_{ab}(q))$ of its component functions at $q$.

Convention. In the four-dimensional case $n=4$ used in this library, indices are $a,b,c,d=0,1,2,3$ with $0$ the time-coordinate label when such a coordinate is chosen, and we write $A_{ab}$ for the components. A covariant $2$-tensor field is also called a $(0,2)$ tensor field.

Convention pointer. The abstract-index reading of the display $A_{ab}=A_q({\mathrm{\partial }}_a{\mathrm{\mid }}_q,{\mathrm{\partial }}_b{\mathrm{\mid }}_q)$ and the signature convention of this library are fixed for the Lorentzian case in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model on this page.