definition of \(C^\infty\) \((2, 0)\)-tensors field induced by Riemannian metric over \(C^\infty\) manifold with boundary
Topics
About: Riemannian manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of \(C^\infty\) \((2, 0)\)-tensors field induced by Riemannian metric over \(C^\infty\) manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( (M, g)\): \(\in \{\text{ the Riemannian manifolds with boundary }\}\)
\( \hat{g}\): \(= \text{ the } 'C^\infty \text{ vectors bundles - } C^\infty \text{ vectors bundle homomorphisms' isomorphism from tangent vectors bundle onto cotangent vectors bundle with respect to } g\)
\( (T^2_0 (TM), M, \pi)\): \(= \text{ the } C^\infty (2, 0) \text{ -tensors bundle over } M\)
\(*\widetilde{g}\): \(: M \to T^2_0 (TM)\), \(\in \{\text{ the } C^\infty (2, 0) \text{ -tensors fields over } M\}\)
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Conditions:
\(\forall m \in M (\widetilde{g} (m): T_mM^* \times T_mM^* \to \mathbb{R}, (t, t') \mapsto t (\hat{g}^{-1} (t')))\)
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With any chart, \((U_m \subseteq M, \phi_m)\), and the induced charts for \(T^2_0 (TM)\), \(\widetilde{g} = \widetilde{g}^{j, l} \partial / \partial x^j \otimes \partial / \partial x^l\), where \(\begin{pmatrix} \widetilde{g}^{j, l} \end{pmatrix}\) is the inverse of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\), as is seen in Note.
The name may seem cumbersome, but at least \(\widetilde{g}\) is not "the inverse" of \(g\), because any inverse of \(g: M \to T^0_2 (TM)\) would have to be \(: T^0_2 (TM) \to M\).
2: Note
Let us see that \(\widetilde{g}\) is indeed well-defined and is indeed a \(C^\infty\) section.
\(\widetilde{g} (m): T_mM^* \times T_mM^* \to \mathbb{R}, (t, t') \mapsto t (\hat{g}^{-1} (t'))\) is multilinear, because \(\widetilde{g} (m) (r_1 t^1 + r_2 t^2, t') = (r_1 t^1 + r_2 t^2) (\hat{g}^{-1} (t')) = r_1 t^1 (\hat{g}^{-1} (t')) + r_2 t^2 (\hat{g}^{-1} (t')) = r_1 \widetilde{g} (m) (t_1, t') + r_2 \widetilde{g} (m) (t_2, t')\); \(\widetilde{g} (m) (t, r'_1 t'_1 + r'_2 t'_2) = t (\hat{g}^{-1} (r'_1 t'_1 + r'_2 t'_2)) = t (r'_1 \hat{g}^{-1} (t'_1) + r'_2 \hat{g}^{-1} (t'_2)) = r'_1 t (\hat{g}^{-1} (t'_1)) + r'_2 t (\hat{g}^{-1} (t'_2)) = r'_1 \widetilde{g} (m) (t, t'_1) + r'_2 \widetilde{g} (m) (t, t'_2)\).
So, \(\widetilde{g} (m) \in T^2_0 (T_mM)\).
So, \(\widetilde{g}\) is a section into \(T^2_0 (TM)\).
For each \(m \in M\), let us take any chart, \((U_m \subseteq M, \phi_m)\) and the induced chart for \(T^2_0 (TM)\).
With the charts, \(\widetilde{g} = \widetilde{g}^{j, l} \partial / \partial x^j \otimes \partial / \partial x^l\).
\(\begin{pmatrix} \widetilde{g}^{j, l} \end{pmatrix}\) is the inverse of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\), because denoting the inverse of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\) as \(\begin{pmatrix} h^{j, l} \end{pmatrix}\), we know that \(\hat{g}^{-1} (t) = h^{j, n} t_n \partial / \partial x^j\) by Note for the definition of '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism from tangent vectors bundle onto cotangent vectors bundle with respect to Riemannian metric, and \(\widetilde{g}^{j, l} = \widetilde{g} (d x^j, d x^l) = d x^j (\hat{g}^{-1} (d x^l)) = d x^j (h^{m, l} \partial / \partial x^m) = h^{m, l} d x^j (\partial / \partial x^m) = h^{m, l} \delta^j_m = h^{j, l}\).
\(\widetilde{g}\) is \(C^\infty\), because as \(\begin{pmatrix} g_{j, l} \end{pmatrix}\) is \(C^\infty\), the inverse, \(\begin{pmatrix} \widetilde{g}^{j, l} \end{pmatrix}\), is \(C^\infty\): by the Laplace expansion, the components of \(\begin{pmatrix} \widetilde{g}^{j, l} \end{pmatrix}\) are some polynomials of the components of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\) divided by the determinant of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\), which is not zero.
