Gauss-Weingarten map - Revision history

https://diffgeom.subwiki.org/w/index.php?action=history&feed=atom&title=Gauss-Weingarten_map Gauss-Weingarten map - Revision history 2026-09-16T12:42:28Z Revision history for this page on the wiki MediaWiki 1.41.2 https://diffgeom.subwiki.org/w/index.php?title=Gauss-Weingarten_map&diff=603&oldid=prev Vipul: 2 revisions 2008-05-18T19:41:19Z

<p>2 revisions</p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <tr class="diff-title" lang="en"> <td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:41, 18 May 2008</td> </tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div> </td></tr></table>

Vipul https://diffgeom.subwiki.org/w/index.php?title=Gauss-Weingarten_map&diff=602&oldid=prev Vipul at 20:30, 3 April 2008 2008-04-03T20:30:27Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:30, 3 April 2008</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td> <td colspan="2" class="diff-lineno">Line 14:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For codimension one oriented submanifolds, we can identify the oriented Grassmannian with the [[sphere]] <math>S^{n-1} = S^m</math>, whereas for codimension one non-oriented submanifolds, we can identify the Grassmannian with [[real projective space]] <math>\R\mathbb{P}^m</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For codimension one oriented submanifolds, we can identify the oriented Grassmannian with the [[sphere]] <math>S^{n-1} = S^m</math>, whereas for codimension one non-oriented submanifolds, we can identify the Grassmannian with [[real projective space]] <math>\R\mathbb{P}^m</math>.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Tangent space can be replaced by normal space===</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We can define the Gauss-Weingarten map, instead, by sending each point to its ''normal space'' (the orthogonal complement of the tangent space to <math>M</math> in the tangent space to <math>\R^n</math>) with suitable orientation, if the manifold is oriented. The two maps are equivalent, under the isomorphism between the Grassmannian of <math>m</math>-dimensional subspaces, and the Grassmannian of <math>(n-m)</math>-dimensional subspaces.</ins></div></td></tr> </table>

Vipul https://diffgeom.subwiki.org/w/index.php?title=Gauss-Weingarten_map&diff=601&oldid=prev Vipul: New page: ==Definition== Suppose <math>M</math> is a differential manifold of dimension <math>m</math>, embedded smoothly inside <math>\R^n</math>. ===For non-oriented submanifolds=== If <mat... 2008-04-03T20:28:49Z

<p>New page: ==Definition== Suppose <math>M</math> is a <a href="/wiki/Differential_manifold" title="Differential manifold">differential manifold</a> of dimension <math>m</math>, embedded smoothly inside <math>\R^n</math>. ===For non-oriented submanifolds=== If <mat...</p> <p><b>New page</b></p><div>==Definition==<br /> <br /> Suppose <math>M</math> is a [[differential manifold]] of dimension <math>m</math>, embedded smoothly inside <math>\R^n</math>.<br /> <br /> ===For non-oriented submanifolds===<br /> <br /> If <math>M</math> is not assumed to have an orientation, the Gauss-Weingarten map is a map from <math>M</math> to the [[Grassmannian manifold]] of <math>m</math>-dimensional subspaces of <math>\R^n</math>, as follows: any point <math>p \in M</math> is mapped to the vector subspace of <math>\R^n</math> parallel to the tangent space <math>T_pM</math>.<br /> <br /> ===For oriented submanifolds===<br /> <br /> If we give an orientation to <math>M</math> the Gauss-Weingarten map is a map from <math>M</math> to the [[oriented Grassmannian manifold]] of <math>m</math>-dimensional subspaces of <math>\R^n</math> as follows: any point <math>p \in M</math> is sent to the vector subspace parallel to the tangent space <math>T_p M</math>, equipped with the orientation.<br /> <br /> ===For codimension one submanifolds===<br /> <br /> For codimension one oriented submanifolds, we can identify the oriented Grassmannian with the [[sphere]] <math>S^{n-1} = S^m</math>, whereas for codimension one non-oriented submanifolds, we can identify the Grassmannian with [[real projective space]] <math>\R\mathbb{P}^m</math>.</div>

Vipul