description/proof of what lowering or raising of index of tensor components is doing for Riemannian manifold with boundary
Topics
About: Riemannian manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \((p, q)\)-tensors space at point on \(C^\infty\) manifold with boundary.
- The reader knows 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.
- The reader knows a definition of \(C^\infty\) \((2, 0)\)-tensors field induced by Riemannian metric over \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any 'vectors spaces - linear morphisms' isomorphism is a map that maps any basis onto a basis bijectively and expands the mapping linearly.
- The reader admits the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.
- The reader admits the proposition that the tensor product of any \(k\) finite-dimensional vectors spaces has the basis that consists of the classes induced by any basis elements.
Target Context
- The reader will have a description and a proof of what the lowering or the raising of any index of any tensor components is doing for any Riemannian 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:
\((M, g)\): \(\in \{\text{ the Riemannian manifolds with boundary }\}\)
\(m\): \(\in M\)
\((U_m \subseteq M, \phi_m)\): \(\in \{\text{ the charts around } m\}\)
\(t\): \(\in T^p_q (T_mM)\), \(= t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes d x^{l_1} \otimes ... \otimes d x^{l_q}\)
\(l_m\): \(: T^p_q (T_mM) \to T^{p - 1}_{q + 1} (T_mM), t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes d x^{l_1} \otimes ... \otimes d x^{l_q} \mapsto t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_{m - 1}} \otimes \partial / \partial x^{j_{m + 1}} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g} (\partial / \partial x^{j_m}) \times d x^{l_1} \otimes ... \otimes d x^{l_q}\)
\(r_m\): \(: T^p_q (T_mM) \to T^{p + 1}_{q - 1} (T_mM), t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes d x^{l_1} \otimes ... \otimes d x^{l_q} \mapsto t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g}^{-1} (d x^{l_m}) \otimes d x^{l_1} \otimes ... \otimes d x^{l_{m - 1}} \otimes d x^{l_{m + 1}} \otimes ... \otimes d x^{l_q}\)
//
Statements:
\(\{g_{j'_m, j_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) represents \(l_m (t)\)
\(\land\)
\(\{\widetilde{g}^{l'_m, l_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) represents \(r_m (t)\)
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2: Note
\(\{g_{j'_m, j_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) looks like in \(T_mM \times ... \times T_mM \times T_mM^* \times T_mM \times ... \times T_mM \times T_mM^* \times ... \times T_mM^*\) instead of in \(T^{p - 1}_{q + 1} (T_mM) = T_mM \times ... \times T_mM \times T_mM^* \times ... \times T_mM^*\), but the 2 spaces are canonically 'vectors spaces - linear morphisms' isomorphic, and \(\{g_{j'_m, j_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) is identified with the corresponding element in \(T^{p - 1}_{q + 1}\).
In fact, \(T_mM \times ... \times T_mM \times T_mM^* \times T_mM \times ... \times T_mM \times T_mM^* \times ... \times T_mM^*\) is not bad in any way, but as we have not given any succinct notation for it, we put \(g_{j'_m, j_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\) in the succinctly denoted \(T^{p - 1}_{q + 1} (T_mM)\).
We do so because where we put that intruding \(T_mM^*\) does not really matter: of course, once the position is defined, it matters that we stick to the definition.
3: Proof
Whole Strategy: Step 1: think of the \(T^1_0 (T_mM)\) case and the \(T^0_1 (T_mM)\) case; Step 2: think of the general \(T^p_q (T_mM)\) case; Step 3: see that the \(T^1_0 (T_mM)\) case and the \(T^0_1 (T_mM)\) case conform to the general case.
Step 1:
Let us see that the \(T^1_0 (T_mM)\) case is simple.
\(\{g_{j', j} t^j\}\) represents \(\hat{g} (t)\), because \(\hat{g} (t) = g_{l, j} t^l d x^j = g_{j, l} t^l d x^j\), by 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.
Let us see that the \(T^0_1 (T_mM)\) case is simple.
\(\{\widetilde{g}^{j', j} t_j\}\) represents \(\hat{g}^{-1} (t)\), because \(\hat{g}^{-1} (t) = \widetilde{g}^{j, l} t_l \partial / \partial x^j\), by 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.
Step 2:
Let us think of the general \(T^p_q (T_mM)\) case.
The issue for the general case is that each of \(\hat{g}\) and \(\hat{g}^{-1}\) cannot be (at least directly) applied to \(t\).
So, what is the index-lowered or index-raised object?, which is the concern of this proposition.
As \(\hat{g}\) is a 'vectors spaces - linear morphisms' isomorphism, it maps the basis for \(T_mM\), \(\{\partial / \partial x^{j_m}\}\), onto a basis for \(T_mM^*\), by the proposition that any 'vectors spaces - linear morphisms' isomorphism is a map that maps any basis onto a basis bijectively and expands the mapping linearly, so, \(\{\hat{g} (\partial / \partial x^{j_m})\}\) is a basis for \(T_mM^*\).
So, \(\{\partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_{m - 1}} \otimes \partial / \partial x^{j_{m + 1}} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g} (\partial / \partial x^{j_m}) \times d x^{l_1} \otimes ... \otimes d x^{l_q}\}\) is a basis for \(T^{p - 1}_{q + 1} (T_mM)\), by the proposition that the tensor product of any \(k\) finite-dimensional vectors spaces has the basis that consists of the classes induced by any basis elements: that proposition does not require the bases for \(V_j\) s to be some specific bases like being the standard bases.
\(l_m\) is a linear expansion of the mapping of the basis.
So, \(l_m\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.
Let us see that \(\{g_{j'_m, j_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) indeed represents \(l_m (t)\).
\(l_m (t) = t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_{m - 1}} \otimes \partial / \partial x^{j_{m + 1}} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g} (\partial / \partial x^{j_m}) \times d x^{l_1} \otimes ... \otimes d x^{l_q} = t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_{m - 1}} \otimes \partial / \partial x^{j_{m + 1}} \otimes ... \otimes \partial / \partial x^{j_p} \otimes (g_{j_m, j'_m} d x^{j'_m}) \times d x^{l_1} \otimes ... \otimes d x^{l_q} = g_{j_m, j'_m} t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_{m - 1}} \otimes \partial / \partial x^{j_{m + 1}} \otimes ... \otimes \partial / \partial x^{j_p} \otimes d x^{j'_m} \times d x^{l_1} \otimes ... \otimes d x^{l_q}\).
That means that \(\{g_{j_m, j'_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) are the components of \(l_m (t)\).
The situation for \(r_m\) is likewise as follows.
As \(\hat{g}^{-1}\) is a 'vectors spaces - linear morphisms' isomorphism, it maps the basis for \(T_mM^*\), \(\{d x^{j_m}\}\), onto a basis for \(T_mM\), by the proposition that any 'vectors spaces - linear morphisms' isomorphism is a map that maps any basis onto a basis bijectively and expands the mapping linearly, so, \(\{\hat{g}^{-1} (d x^{j_m})\}\) is a basis for \(T_mM\).
So, \(\partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g}^{-1} (d x^{l_m}) \otimes d x^{l_1} \otimes ... \otimes d x^{l_{m - 1}} \otimes d x^{l_{m + 1}} \otimes ... \otimes d x^{l_q}\) is a basis for \(T^{p + 1}_{q - 1} (T_mM)\), by the proposition that the tensor product of any \(k\) finite-dimensional vectors spaces has the basis that consists of the classes induced by any basis elements.
\(r_m\) is a linear expansion of the mapping of the basis.
So, \(r_m\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.
Let us see that \(\{\widetilde{g}^{l'_m, l_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) indeed represents \(r_m (t)\).
\(r_m (t) = t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \hat{g}^{-1} (d x^{l_m}) \otimes d x^{l_1} \otimes ... \otimes d x^{l_{m - 1}} \otimes d x^{l_{m + 1}} \otimes ... \otimes d x^{l_q} = t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes (\widetilde{g}^{l'_m, l_m} \partial / \partial x^{l'_m}) \otimes d x^{l_1} \otimes ... \otimes d x^{l_{m - 1}} \otimes d x^{l_{m + 1}} \otimes ... \otimes d x^{l_q} = \widetilde{g}^{l'_m, l_m} t^{j_1, ..., j_p}_{l_1, ..., l_q} \partial / \partial x^{j_1} \otimes ... \otimes \partial / \partial x^{j_p} \otimes \partial / \partial x^{l'_m} \otimes d x^{l_1} \otimes ... \otimes d x^{l_{m - 1}} \otimes d x^{l_{m + 1}} \otimes ... \otimes d x^{l_q}\).
That means that \(\{\widetilde{g}^{l'_m, l_m} t^{j_1, ..., j_p}_{l_1, ..., l_q}\}\) are the components of \(r_m (t)\).
Step 3:
Let us see that the \(T^1_0 (T_mM)\) case and the \(T^0_1 (T_mM)\) case conform to the general case (in fact, they should logically, but let us see it explicitly).
\(\hat{g}\) is \(t = t^j \partial / \partial x^j \mapsto g_{j, l} t^l d x^j = g_{l, j} t^j d x^l = t^j g_{l, j} d x^l = t^j \hat{g} (\partial / \partial x^j)\), which is what the general case dictates.
\(\hat{g}^{-1}\) is \(t = t_j d x^j \mapsto \widetilde{g}^{j, l} t_l \partial / \partial x^j = \widetilde{g}^{l, j} t_j \partial / \partial x^l = t_j \widetilde{g}^{l, j} \partial / \partial x^l = t_j \hat{g}^{-1} (d x^j)\), which is what the general case dictates.
