{"id":31233,"date":"2016-10-17T00:52:04","date_gmt":"2016-10-17T05:52:04","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=31233"},"modified":"2016-10-17T06:23:53","modified_gmt":"2016-10-17T11:23:53","slug":"manifold-26","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/17\/manifold-26\/","title":{"rendered":"What is a Manifold? (2\/6)"},"content":{"rendered":"<p>We continue from <a href=\"http:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/07\/manifold-article-16\/\">Part One of this journey<\/a>\u00a0our attempt to illustrate how one can start with calculus and arrive at the definition of a 1-dimensional manifold.<\/p>\n<p>In the previous segment, we concluded with the fact that a curve in <img loading=\"lazy\" decoding=\"async\" class=\"latex\" title=\"\\mathbb{R}^3\" src=\"http:\/\/s0.wp.com\/latex.php?zoom=1.5&amp;latex=%5Cmathbb%7BR%7D%5E3&amp;bg=ffffff&amp;fg=000&amp;s=0\" alt=\"\\mathbb{R}^3\" width=\"18\" height=\"14\" \/>\u00a0may be viewed as an interval\u00a0<em>I<\/em> together with the following set of data:<\/p>\n<ul>\n<li>\u00a0A real-valued function <img loading=\"lazy\" decoding=\"async\" class=\"latex\" title=\"\\ell\u00a0(t)\" src=\"http:\/\/s0.wp.com\/latex.php?zoom=1.5&amp;latex=%5Cell%C2%A0%28t%29&amp;bg=ffffff&amp;fg=000&amp;s=0\" alt=\"\\ell\u00a0(t)\" width=\"24\" height=\"18\" \/>, which will help measure the length,<\/li>\n<li>A real-valued function <img loading=\"lazy\" decoding=\"async\" class=\"latex\" title=\"k(t)\" src=\"http:\/\/s0.wp.com\/latex.php?zoom=1.5&amp;latex=k%28t%29&amp;bg=ffffff&amp;fg=000&amp;s=0\" alt=\"k(t)\" width=\"26\" height=\"18\" \/>, which will help measure the curvature, and,<\/li>\n<li>A real-valued function <img loading=\"lazy\" decoding=\"async\" class=\"latex\" title=\"\\tau (t)\" src=\"http:\/\/s0.wp.com\/latex.php?zoom=1.5&amp;latex=%5Ctau+%28t%29&amp;bg=ffffff&amp;fg=000&amp;s=0\" alt=\"\\tau (t)\" width=\"26\" height=\"18\" \/>, which will help measure the torsion.<\/li>\n<\/ul>\n<p>In this segment we will see some examples and discuss possible constructions that this allows us.<\/p>\n<p><!--more--><\/p>\n<p><strong>Examples:<\/strong><\/p>\n<p>The data\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28I%2C%5Cell%2C+k%2C+%5Ctau%29+%3D%28%5B0%2C1%5D%2C+%5C+%5Csqrt%7B1%2B4t%5E2%7D%2C+%5C+2%281%2B4t%5E2%29%5E%7B-3%2F2%7D%2C+%5C+0%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"(I,&#92;ell, k, &#92;tau) =([0,1], &#92; &#92;sqrt{1+4t^2}, &#92; 2(1+4t^2)^{-3\/2}, &#92; 0)\" class=\"latex\" \/> represents the\u00a0part of the parabola <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3Dx%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"y=x^2\" class=\"latex\" \/> between <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=0+%5Cleq+x+%5Cleq+1&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"0 &#92;leq x &#92;leq 1\" class=\"latex\" \/>. More precisely, this segment of the parabola is the only\u00a0curve with the above\u00a0curvature and torsion; any other curve is just the same one with a change in the position of the observer in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D%5E3&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}^3\" class=\"latex\" \/>.<\/p>\n<p>The data\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28I%2C%5Cell%2C+k%2C+%5Ctau%29+%3D%28%5B0%2C%5Cinfty%29%2C+%5Csqrt%7B2%7D%2C+0.5%2C+0.5%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"(I,&#92;ell, k, &#92;tau) =([0,&#92;infty), &#92;sqrt{2}, 0.5, 0.5)\" class=\"latex\" \/> represents\u00a0the helix <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28%5Ccos+%28t%29%2C+%5Csin+%28t%29%2C+t%29.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"(&#92;cos (t), &#92;sin (t), t).\" class=\"latex\" \/><\/p>\n<p><a href=\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/330px-Helix.svg_.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-31299 aligncenter\" src=\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/330px-Helix.svg_-300x219.png\" alt=\"330px-helix-svg\" width=\"300\" height=\"219\" srcset=\"https:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/330px-Helix.svg_-300x219.png 300w, https:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/330px-Helix.svg_.png 754w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p><strong>Equivalence of Curves<\/strong><\/p>\n<p>Often,\u00a0the first thing one does after defining\/constructing a new object in math is to define an appropriate notion of &#8220;sameness&#8221; between two of them. To this end, we have the following definition:<\/p>\n<p><em>Definition<\/em><strong>:<\/strong>\u00a0Two curves\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28I_1%2C+%5Cell+_1%2C+k_1%2C+%5Ctau+_1%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"(I_1, &#92;ell _1, k_1, &#92;tau _1)\" class=\"latex\" \/> and\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28I_2%2C+%5Cell+_2%2C+k_2%2C+%5Ctau+_2%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"(I_2, &#92;ell _2, k_2, &#92;tau _2)\" class=\"latex\" \/> are equivalent if there exists a diffeomorphism <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%3A+I_1+%5Clongrightarrow+I_2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;phi: I_1 &#92;longrightarrow I_2\" class=\"latex\" \/> such that\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cforall+t%2C&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;forall t,\" class=\"latex\" \/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cell_1%28t%29+%3D+%5Cell+_2+%28%5Cphi+%28t%29%29+%7C%5Cphi+%27+%28t%29%7C.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;ell_1(t) = &#92;ell _2 (&#92;phi (t)) |&#92;phi &#039; (t)|.\" class=\"latex\" \/><\/p>\n<p style=\"text-align: left\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=k_1%28t%29%3Dk_2%28%5Cphi+%28t%29%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"k_1(t)=k_2(&#92;phi (t))\" class=\"latex\" \/><\/p>\n<p style=\"text-align: left\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau%C2%A0_1%28t%29%3D%5Ctau%C2%A0_2%28%5Cphi+%28t%29%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau\u00a0_1(t)=&#92;tau\u00a0_2(&#92;phi (t))\" class=\"latex\" \/><\/p>\n<p>In calculus language, this would be a &#8220;re-parameterization&#8221; of a curve. If this happens, then the\u00a0two curves that they describe\u00a0coincide completely &#8212; they will overlap\u00a0after a rotation and translation.<\/p>\n<p>The above definition is derived from the plausible requirement for the following to hold:<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cint+_%7Bc%7D%5Ed+f%28t%29+%5Cell+_1%28t%29%C2%A0dt+%3D%C2%A0%5Cint+_%7B%5Cphi+%28c%29%7D%5E%7B%5Cphi+%28d%29%7D+f%28s%29+%5Cell+_2%28s%29%C2%A0ds%2C+%5C+%5Cforall+%5Bc%2C+d%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;int _{c}^d f(t) &#92;ell _1(t)\u00a0dt =\u00a0&#92;int _{&#92;phi (c)}^{&#92;phi (d)} f(s) &#92;ell _2(s)\u00a0ds, &#92; &#92;forall [c, d]\" class=\"latex\" \/><\/p>\n<p>A change of variable <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=s%3D%5Cphi+%28u%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"s=&#92;phi (u)\" class=\"latex\" \/> in the second integral leads to the first condition in the definition above.<\/p>\n<div id=\"attachment_31300\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/equi.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-31300\" class=\"wp-image-31300 size-medium\" src=\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/equi-300x249.png\" width=\"300\" height=\"249\" srcset=\"https:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/equi-300x249.png 300w, https:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/10\/equi.png 690w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-31300\" class=\"wp-caption-text\">Image by Behnam Esmayli.<\/p><\/div>\n<p><strong>Example<\/strong>:<\/p>\n<p>We saw from our previous example that <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28%5B0%2C1%5D%2C+%5C+%C2%A0%5Csqrt%7B1%2B4t%5E2%7D%2C+%5C+2%281%2B4t%5E2%29%5E%7B-3%2F2%7D%2C+0%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"([0,1], &#92; \u00a0&#92;sqrt{1+4t^2}, &#92; 2(1+4t^2)^{-3\/2}, 0)\" class=\"latex\" \/> describes a part of the parabola <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=y%3Dx%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"y=x^2\" class=\"latex\" \/>.<\/p>\n<p>It turns out that the set of data <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%28%5B0%2C1%5D%2C+%5C+%5Csqrt%7B1%2B%5Cfrac+%7B4%7D%7Bs%7D%7D%2C+%5C+2%281%2B4s%29%5E%7B-3%2F2%7D%2C+%5C+0%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"([0,1], &#92; &#92;sqrt{1+&#92;frac {4}{s}}, &#92; 2(1+4s)^{-3\/2}, &#92; 0)\" class=\"latex\" \/> also describes the same curve, with an explicit equivalence being\u00a0given by the map <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=s%3Dt%5E2%3D%5Cphi+%28t%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"s=t^2=&#92;phi (t)\" class=\"latex\" \/>.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Intrinsic Integration On Curves<\/strong><\/p>\n<p>Suppose that we are given a function which assigns to each point on\u00a0a curve\u00a0a real number, say the temperature at that\u00a0point. We wish to find the average temperature over the curve. To do this, we need to add up (integrate) the temperatures at each point and divide the result by total number of points (length).<\/p>\n<p>The following is an appropriate definition of the integral of a real-valued continuous function <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f\" class=\"latex\" \/> on our manifold:<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cint+f+%3A%3D+%5Cint+_%7Bt_1%7D%5E%7Bt_2%7D+f%28t%29+%5Cell+%28t%29%C2%A0dt.+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;int f := &#92;int _{t_1}^{t_2} f(t) &#92;ell (t)\u00a0dt. \" class=\"latex\" \/><\/p>\n<p>Again, it is adding up values of <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f\" class=\"latex\" \/> on <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=I&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"I\" class=\"latex\" \/>, just as in Riemann sum, except that now we have a non-uniform weight <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cell+%28t%29+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;ell (t) \" class=\"latex\" \/> applied to each summand.<\/p>\n<p>The reason this definition works is that if we calculate the integral using a different but\u00a0equivalent set of data, then by the change-of-variables formula, we would get the same answer:<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cint+f+%3A%3D+%5Cint+_%7Bt_1%7D%5E%7Bt_2%7D+f%28t%29+%5Cell+%28t%29%C2%A0dt+%3D+%C2%A0%5Cint+_%7Bs_1%7D%5E%7Bs_2%7D+f%28s%29+%5Cell+%28s%29%C2%A0ds.+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;int f := &#92;int _{t_1}^{t_2} f(t) &#92;ell (t)\u00a0dt = \u00a0&#92;int _{s_1}^{s_2} f(s) &#92;ell (s)\u00a0ds. \" class=\"latex\" \/><\/p>\n<p><em>Observation: <\/em>Our definition of integration only depended on the function <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cell&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;ell\" class=\"latex\" \/>. Therefore, it suffices to have a way of measuring distances in order to be able to define a parameterization-free, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D%5E3&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}^3\" class=\"latex\" \/>-free notion of integration on a curve.<\/p>\n<p><em>Note:\u00a0<\/em>The integration above may be familiar from\u00a0calculus, or complex functions, as a &#8220;line integral&#8221;. But it is usually not emphasized there that this is intrinsic to the curve. However, it indeed comes with the curve, independent of the embedding of the curve in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D%5E3.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}^3.\" class=\"latex\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>What&#8217;s next &#8230;<\/strong><\/p>\n<p>Now that we know a way of telling when two parameterizations coincide, in the next installment, we will be able to &#8220;glue together&#8221; pieces to arrive at a manifold.<\/p>\n<p>The definition of the integration will work globally because different parameterizations\u00a0agree on it locally.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>We continue from Part One of this journey\u00a0our attempt to illustrate how one can start with calculus and arrive at the definition of a 1-dimensional manifold. In the previous segment, we concluded with the fact that a curve in \u00a0may &hellip; <a href=\"https:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/17\/manifold-26\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/17\/manifold-26\/><\/div>\n","protected":false},"author":118,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[12],"tags":[],"class_list":["post-31233","post","type-post","status-publish","format-standard","hentry","category-math"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3gbww-87L","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/31233","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/users\/118"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/comments?post=31233"}],"version-history":[{"count":23,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/31233\/revisions"}],"predecessor-version":[{"id":31302,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/31233\/revisions\/31302"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/media?parent=31233"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/categories?post=31233"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/tags?post=31233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}