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kappaka
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I'm reading a paper and there is a proof that the double of a compact locally conformally flat Riemannian manifold with totally geodesic boundary again carries a locally conformally flat structure. The proof is as follows:
Let \( (M^n, g) \) be a locally conformally flat compact manifold with totally geodesic boundary. We denote a boundary neighborhood in \( M \) by \( U_a \cup \partial U_a \), where \( U_a \) is open and \( \partial U_a = \partial M \cap \partial U_a \) is a segment on the boundary. By definition, there is a conformal map \( \varphi_a : U_a \cup \partial U_a \to V_a \cup \partial V_a \subset S^n_+ \cup S^{n-1} \), such that \( V_a \subset S^n_+ \) and \( \partial V_a \) is on the equator. Denote the doubling of \( M \) by \( N = M \cup M^* \). We will define a locally conformally flat smooth structure on \( N \).
Define the corresponding conformal map \( \varphi^*_a \) from \( U^*_a \subset M^* \) to \( V^*_a \subset S^n_- \) through reflection. If \( \varphi_b \) and \( \varphi^*_b \) are another pair of conformal maps such that \( U_a \cap U_b \) (and thus \( U^*_a \cap U^*_b \)) is nonempty, then there is a conformal transformation from \( \varphi_a(U_a \cap U_b) \) to \( \varphi_b(U_a \cap U_b) \). Similarly, there is a corresponding conformal transformation \( \varphi^* \) on the counterpart.
By Liouville's theorem, the conformal transformations \( \varphi \) and \( \varphi^* \) can be extended to conformal transformations on \( S^n \), still denoted by \( \varphi \) and \( \varphi^* \). If we can prove that \( \varphi = \varphi^* \), then they define a locally conformally flat smooth structure on \( N \).
Suppose that \( \varphi \) and \( \varphi^* \) are not equal. Then \( \varphi^{-1} \circ \varphi^* \) is not the identity map on \( S^n \). Notice that it is the identity map on \( \varphi_a(\partial(U_a \cap U_b)) \), which is a co-dimension one submanifold contained in the equator. Thus, \( \varphi^{-1} \circ \varphi^* \) must be a reflection with respect to the equator (see, for example, Chap. A in [1]). This gives us a contradiction because \( \varphi^{-1} \) cannot map \( \varphi^*_b(U^*_a \cap U^*_b) \subset S^n_- \) to \( \varphi_a(U_a \cap U_b) \subset S^n_+ \).
I'm getting lost on their definition of the conformal boundary map. What is this set \( U_a \cup \partial U_a \)? Is it even open? Don’t we have to find an open set that covers the points where \( M \) is attached to itself, i.e., something like a tubular neighborhood of the boundary, which we then mirror?
Could someone elaborate?
Let \( (M^n, g) \) be a locally conformally flat compact manifold with totally geodesic boundary. We denote a boundary neighborhood in \( M \) by \( U_a \cup \partial U_a \), where \( U_a \) is open and \( \partial U_a = \partial M \cap \partial U_a \) is a segment on the boundary. By definition, there is a conformal map \( \varphi_a : U_a \cup \partial U_a \to V_a \cup \partial V_a \subset S^n_+ \cup S^{n-1} \), such that \( V_a \subset S^n_+ \) and \( \partial V_a \) is on the equator. Denote the doubling of \( M \) by \( N = M \cup M^* \). We will define a locally conformally flat smooth structure on \( N \).
Define the corresponding conformal map \( \varphi^*_a \) from \( U^*_a \subset M^* \) to \( V^*_a \subset S^n_- \) through reflection. If \( \varphi_b \) and \( \varphi^*_b \) are another pair of conformal maps such that \( U_a \cap U_b \) (and thus \( U^*_a \cap U^*_b \)) is nonempty, then there is a conformal transformation from \( \varphi_a(U_a \cap U_b) \) to \( \varphi_b(U_a \cap U_b) \). Similarly, there is a corresponding conformal transformation \( \varphi^* \) on the counterpart.
By Liouville's theorem, the conformal transformations \( \varphi \) and \( \varphi^* \) can be extended to conformal transformations on \( S^n \), still denoted by \( \varphi \) and \( \varphi^* \). If we can prove that \( \varphi = \varphi^* \), then they define a locally conformally flat smooth structure on \( N \).
Suppose that \( \varphi \) and \( \varphi^* \) are not equal. Then \( \varphi^{-1} \circ \varphi^* \) is not the identity map on \( S^n \). Notice that it is the identity map on \( \varphi_a(\partial(U_a \cap U_b)) \), which is a co-dimension one submanifold contained in the equator. Thus, \( \varphi^{-1} \circ \varphi^* \) must be a reflection with respect to the equator (see, for example, Chap. A in [1]). This gives us a contradiction because \( \varphi^{-1} \) cannot map \( \varphi^*_b(U^*_a \cap U^*_b) \subset S^n_- \) to \( \varphi_a(U_a \cap U_b) \subset S^n_+ \).
I'm getting lost on their definition of the conformal boundary map. What is this set \( U_a \cup \partial U_a \)? Is it even open? Don’t we have to find an open set that covers the points where \( M \) is attached to itself, i.e., something like a tubular neighborhood of the boundary, which we then mirror?
Could someone elaborate?
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