{"id":109,"date":"2024-07-22T17:32:48","date_gmt":"2024-07-22T17:32:48","guid":{"rendered":"https:\/\/summergeometry.org\/sgi2024\/?p=109"},"modified":"2024-07-26T22:46:19","modified_gmt":"2024-07-26T22:46:19","slug":"bridging-curvatures-from-riemann-to-gauss","status":"publish","type":"post","link":"https:\/\/summergeometry.org\/sgi2024\/bridging-curvatures-from-riemann-to-gauss\/","title":{"rendered":"Bridging Curvatures: From Riemann to Gauss"},"content":{"rendered":"\n

With a discussion on general relativity by Artur RB Boyago.<\/strong><\/p>\n\n\n\n

Cover image from professor Keenan Crane’s discrete differential geometry note<\/a>.<\/p>\n\n\n\n

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Curvatures<\/p>\n<\/div><\/div>\n\n\n\n

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Two fundamental mathematical approaches to describing a shape (space) \\(M\\) involve addressing the following questions:<\/p>\n\n\n\n

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  1. \ud83e\udd14 (Topological) How many holes are present in \\(M\\)?<\/li>\n\n\n\n
  2. \ud83e\uddd0 (Geometrical) What is the curvature of \\(M\\)?<\/li>\n<\/ol>\n\n\n\n

    To address the first question, our space \\(M\\) is considered solely in terms of its topology without any additional structure. For the second question, however, \\(M\\) is assumed to be a Riemannian manifold<\/strong>\u2014a smooth manifold equipped with a Riemannian metric, which is an inner product defined at each point. This category includes surfaces as we commonly understand them. This note aims to demystify Riemannian curvature, connecting it to the more familiar concept of curvature.<\/p>\n\n\n\n

    … Notoriously Difficult…<\/h2>\n\n\n\n

    Riemannian curvature<\/strong>, a formidable name, frequently appears in literatures when digging into geometry. Fields Medalist C\u00e9dric Villani<\/a> in his Optimal Transport<\/em> said<\/p>\n\n\n\n

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    “it is notoriously difficult, even for specialists, to get some understanding of its meaning.” (p.357)<\/p>\n<\/blockquote>\n\n\n

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    Bernhard Riemann<\/a>.<\/figcaption><\/figure>\n<\/div>\n\n\n

    Riemannian curvature is the father of curvatures we saw in our SGI tutorial and other curvatures. In the most abstract sense, Riemann curvature tensor<\/a> is defined as a \\((0,4)\\) tensor field obtained by lowering the contravariant index of the \\((1,3)\\) tensor field defined as<\/p>\n\n\n\n

    \\begin{aligned} R:\\mathfrak{X}(M)\\times \\mathfrak{X}(M)\\times\\mathfrak{X}(M)&\\to \\mathfrak{X}(M)\\\\ (X,Y,Z)&\\mapsto R(X, Y)Z:=\\nabla_Y \\nabla_X Z-\\nabla_X \\nabla_Y Z-\\nabla_{[X, Y]} Z \\end{aligned}<\/p>\n\n\n\n

    where \\(\\nabla\\) is the unique symmetric metric connection on \\(M\\) called Levi-Civita connection<\/a> and \\([X,Y]\\) is the Lie bracket of vector fields<\/a> on \\(M\\) …<\/p>\n\n\n\n

    So impenetrable<\/mark><\/strong><\/em> \ud83d\ude35\u200d\ud83d\udcab. Why bother using four vector fields to characterize the extent to which a space \\(M\\) is curved?<\/p>\n\n\n\n

    To answer this question, we go to its opposite and try to get some of the common ways giving birth mathematical concepts.<\/p>\n\n\n\n

    What Does It Mean to Say Something Is Flat?<\/h2>\n\n\n\n

    Flat Torus<\/h3>\n\n\n\n

    A quote from prof. Crane’s discrete differential geometry note<\/a>:<\/p>\n\n\n\n

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    “It also helps to see pictures of surfaces with zero mean and Gaussian curvature. Zero-curvature surfaces are so well-studied in mathematics that they have special names. Surfaces with zero Gaussian curvature are called developable surfaces because they can be ‘developed’ or flattened out into the plane without any stretching or tearing<\/mark>. For instance, any piece of a cylinder is developable since one of the principal curvatures is zero:” (p.39)<\/p>\n<\/blockquote>\n\n\n

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    \"\"<\/figure>\n<\/div>\n\n\n

    Te following video continues with this cylinder example to show difficulties characterizing flatness. It examines the following question: how to use a piece of flat white paper to roll into a doughnut-shaped object (Torus<\/a>) without distorting the length of the paper<\/mark>. That is, we are not allowed to use a piece of stretchable rubber to do this job.<\/p>\n\n\n\n

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