195 Type-(0,2) Tensor Field

A smooth mapping that assigns a bilinear form on tangent vectors at each point that is used to represent a metric field.

definition [d] (Type-\((0,2)\) Tensor Field = Covariant Rank-2 Tensor Field = Second-Rank Covariant Tensor Field) A smooth section of the tensor bundle \(T^{0}_{\ 2}(M)\): at each point \(P\), a bilinear map

• \(T_{P} : T_{P}M \times T_{P}M \rightarrow \mathbb{R}\) ,

with components that are smooth functions of the coordinates.

where

• \(M\) is a smooth manifold.
• \(P\) is a point of \(M\).
• \(T_{P}\) is the value of the tensor field at \(P\).
• \(T_{P}M\) is the tangent space at \(P\).
• \(T^{0}_{\ 2}(M)\) is the bundle of type-\((0,2)\) tensors on \(M\).
• \(\mathbb{R}\) is the set of real numbers.

Note:

• type \((0,2)\) means two covariant indices and no contravariant indices.
• equivalently written \(\sum_{i,j} T_{ij}\, du^{i}\otimes du^{j}\) in local coframes.

definition [d] (Type-\((0,2)\) Tensor Field = Field of Bilinear Forms) A smooth type-\((0,2)\) tensor field, with two important special cases:

• if \(T_{ij}\) is symmetric and nondegenerate, it is a metric tensor .
• if \(T_{ij}\) is totally antisymmetric, it is a differential \(2\)-form .

where

• \(T_{ij}\) are the components of the type-\((0,2)\) tensor field.

Note:

• the metric is the fundamental physical example of a symmetric \((0,2)\) field.
• nondegeneracy means \(\det(T_{ij}) \neq 0\).

195.1 Elementary Example

195.1.1 Simple

A type-\((0,2)\) tensor field assigns a bilinear form to each point.

\[ U = \{ p,\ q,\ r \} \]

\[ T_{p}(e_{1},e_{1}) = 1,\quad T_{p}(e_{1},e_{2}) = 0 \]

where

• \(T_{p}\) is the bilinear form at the point \(p\).
• \(e_{1}, e_{2}\) are tangent vectors at \(p\).

195.1.2 General

At each point the value is a \(3 \times 3\) matrix of components, as for a metric field.

\[ U = \{ p,\ q \} \]

\[ T_{p} = I_{3},\quad T_{q} = \operatorname{diag}(2,1,1) \]

where

• \(T_{p}\) is the type-\((0,2)\) tensor at \(p\).
• \(I_{3}\) is the \(3 \times 3\) identity matrix.

195.2 References

1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — type-\((0,2)\) tensor fields as sections of \(T^{0}_{2}(M)\).
2. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — covariant \(2\)-tensor fields; metric example.
3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — second-rank covariant tensors.

• Alternating Function
• Alternating k-linear function
• Boundary Operation
• Bundle
• Cartesian Product
• Constant Metric Tensor
• Contravariant Rank
• Covariant Rank
• Covariant, Contravariant, and Coordinate Systems
• Differential Forms
• Differential k-Form
• Edge of a Domain
• Exterior Algebra and Exterior Derivatives
• Forms
• Functions
• Inverse Metric
• k-Covector
• k-Covector Field
• k-Fold Product
• k-form
• k-Tensor
• Lie
• Lie Algebra
• Line Element
• Linear Functional
• Lorentzian Manifold
• Metric Tensor
• Multilinear Function
• Nondegenerate
• Nondegenerate Bilinear Form
• One-Form
• Smooth
• Smooth Assignment
• Smooth Mapping
• Stoke’s Theorem and the Fundamental Theorem of Calculus
• Symmetric Array
• Tangent Space
• Tensor Field
• Tensors
• Two-Form
• Type-(0,2) Tensor Field
• Type-(q,r) Tensor