definition of '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism from tangent vectors bundle onto cotangent vectors bundle w.r.t. Riemannian metric
Topics
About: Riemannian manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of cotangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of Riemannian manifold with boundary.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a 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.
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:
\( (M, g)\): \(\in \{\text{ the Riemannian manifolds with boundary }\}\)
\( (TM, M, \pi)\): \(= \text{ the tangent vectors bundle }\)
\( (TM^*, M, \pi^*)\): \(= \text{ the cotangent vectors bundle }\)
\(*\hat{g}\): \(: TM \to TM^*, v \mapsto g (v, \bullet)\), \(\in \{\text{ the '} C^\infty \text{ vectors bundles - } C^\infty \text{ vectors bundle homomorphisms' isomorphisms } \}\)
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Conditions:
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With any chart, \((U_m \subseteq M, \phi_m)\), and the induced charts for \(TM\), \(TM^*\), and \(T^0_2 (TM)\), \(\hat{g} (v) = g_{l, j} v^l d x^j\), as is seen in Note.
With the charts, \(\hat{g}^{-1} (t) = \widetilde{g}^{j, n} t_n \partial / \partial x^j\), where \(\begin{pmatrix} \widetilde{g}^{j, n} \end{pmatrix}\) is the inverse matrix of \(\begin{pmatrix} g_{j, n} \end{pmatrix}\), as is seen in Note, while \(\widetilde{g}\) is really the \(C^\infty\) \((2, 0)\)-tensors field induced by \(g\) and \(\widetilde{g}^{j, n}\) are its components with respect to the induced chart for \(T^2_0 (TM)\).
2: Note
Let us see that \(\hat{g}\) is indeed well-defined and is indeed a '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism.
Is \(g (v, \bullet)\) indeed in \(TM^*\)?
It is certainly a multilinear map, \(: TM \to \mathbb{R}\), because \(g\) is multilinear \(: TM \times TM \to \mathbb{R}\). So, \(g (v, \bullet) \in \biguplus_{m \in M} L (T_mM: \mathbb{R})\).
\(\biguplus_{m \in M} L (T_mM: \mathbb{R})\) is identified with \(T^0_1 (TM) = TM^*\), by the identification mentioned in the definition of \(C^\infty\) (p, q)-tensors bundle over \(C^\infty \) manifold with boundary.
So, \(g (v, \bullet) \in TM^*\), by the identification.
\(\hat{g}\) is fiber-preserving.
\(\hat{g}\) is linear on each fiber, \(T_mM\): \(g (r v + r' v', \bullet) = r g (v, \bullet) + r' g (v', \bullet) = r \hat{g} (v) + r' \hat{g} (v')\).
\(\hat{g}\) is injective on each fiber: let \(v, v' \in T_mM\) be any such that \(v \neq v'\), let us suppose that \(\hat{g} (v) = g (v, \bullet) = g (v', \bullet) = \hat{g} (v')\); \(0 = g (v, \bullet) - g (v', \bullet) = g (v - v', \bullet)\); especially, \(g (v - v', v - v') = 0\), which would imply \(v - v' = 0\), by the positive-definite-ness of \(g\), a contradiction.
\(\hat{g}\) is bijective on each fiber, by the proposition that any linear injection between any same-finite-dimensional vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
\(\hat{g}\) maps \(T_mM\) onto \(T_mM^*\).
So, \(\hat{g}\) is a bijection.
Let us see that \(\hat{g}\) is \(C^\infty\).
For each \(m \in M\), let us take any chart, \((U_m \subseteq M, \phi_m)\) and the induced charts for \(TM\), \(TM^*\), and \(T^0_2 (TM)\).
With the charts, \(v = v^j \partial / \partial x^j\) and \(g = g_{l, m} d x^l \otimes d x^m\), and \(\hat{g} (v) = g (v, \bullet) = g_{l, m} d x^l \otimes d x^m (v^j \partial / \partial x^j) = g_{l, m} v^j d x^l (\partial / \partial x^j) d x^m = g_{l, m} v^j \delta^l_j d x^m = g_{l, m} v^l d x^m\).
So, the components function of \(\hat{g}\) is \((v^j, x^j) \mapsto (g_{l, j} v^l, x^j)\), which is \(C^\infty\) because \(g\) is \(C^\infty\).
So, \(\hat{g}\) is \(C^\infty\).
So, \(\hat{g}\) is a \(C^\infty\) vectors bundles homomorphism.
By the proposition that for any 2 \(C^\infty\) vectors bundles over any same \(C^\infty\) manifold with boundary, any bijective \(C^\infty\) vectors bundle homomorphism is a '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism, \(\hat{g}\) is a '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism.
With the charts, \(\hat{g}^{-1} (t) = \widetilde{g}^{j, n} t_n \partial / \partial x^j\), where \(\begin{pmatrix} \widetilde{g}^{j, l} \end{pmatrix}\) is the inverse of \(\begin{pmatrix} g_{j, l} \end{pmatrix}\), which exists because \(\begin{pmatrix} g_{j, l} \end{pmatrix}\) is positive definite, because \(\hat{g} (\widetilde{g}^{j, n} t_n \partial / \partial x^j) = g_{l, m} \widetilde{g}^{l, n} t_n d x^m = g_{m, l} \widetilde{g}^{l, n} t_n d x^m = \delta^n_m t_n d x^m = t_m d x^m = t\).
