Riemannian Structures and Triangulations of Manifolds
Riemannian Structures and Triangulations of Manifolds
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黎曼结构和流形三角剖分
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发表时间：
2010
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通讯作者：
V. K. Patodi
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作者：
V. K. Patodi
设X是维数为N的闭C°°-三角形。然后X上的两个额外结构已经被相当广泛地研究。一个是黎曼结构产生黎曼几何，另一个是X的三角剖分产生多面体或组合拓扑。有时我们会遇到这样一个问题，我们有一个流形的不变量，它有一个很好的表达式，用其中一个结构来表示，我们想用另一个结构来表示这个不变量。让我们考虑两个例子。我们的第一个例子与流形X的Pontrjagin类pi 9 1 g i g N/4有关。如果我们在X上选择一个黎曼结构，那么通过著名的Chern-Weil理论（见[2]），我们可以明确地写出一个表示第i个Pontrjagin类的闭4/-形式，它被认为是H（Z，R）的一个元素。得到Pontrjagin类的组合公式是一个公开问题。现在我们来看第二个例子。第二个例子是Reidemeister-Franz挠。设%是%（X）的正交n × n矩阵表示，n是固定的正整数。设ε x是由该表示定义的X上的平坦向量丛。设X的所有系数在向量丛ε z中的上同调群均为零。然后可以定义一个不变量，称为Reidemeister-Franz挠率（参见[3]和[4]），取决于X和%。为了定义这种扭转首先选择一个光滑的三角形的X和在这个三角形的一个定义一个不变的证明是不变的，如果我们考虑细分的三角形。到目前为止，我们还没有一个不使用三角剖分选择的挠率定义。这是一个有趣的公开问题，定义这个不变量的黎曼结构。在这个方向上有一个Ray和Singer的猜想（见[4]）。他们定义了一个不变量，称之为解析挠率。Ray-Singer解析挠率是
Let X be a closed C°°-rrianifold of dimension N. Then two additional structures on X have been quite extensively studied. One is the Riemannian structure giving rise to Riemannian geometry and the other one is the triangulation of X giving rise to polyhedral or combinatorial topology. Sometimes we come across a problem in which we have an invariant for the manifold which has a nice expression in terms of one of these structures and we want to express this invariant in terms of the other structure. Let us consider two examples. Our first example is related to Pontrjagin classes pi9 1 g i g N/4, of the manifold X. If we choose a Riemannian structure on X, then by the well-known Chern-Weil theory (see [2]) we can explicitly write down a closed 4/-form representing the ith Pontrjagin class regarded as an element of H(Z, R). It is an open problem to obtain a combinatorial formula for Pontrjagin classes. We now come to our second example. Our second example refers to Reidemeister-Franz torsion. Let % be a representation of %{X) by orthogonal n x n matrices, n a fixed positive integer. Let £x be the flat vector bundle on X defined by this representation. Suppose that all the cohomology groups of X with coefficients in the vector bundle £z are zero. Then one can define an invariant known as Reidemeister-Franz torsion (see [3] and [4]) depending on X and %. To define this torsion one first chooses a smooth triangulation of X and in terms of this triangulation one defines an invariant which one proves is not changed if we consider subdivisions of the triangulation. We do not have so far a definition for this torsion which does not make use of the choice of a triangulation. It is an interesting open problem to define this invariant in terms of a Riemannian structure. There is a conjecture of Ray and Singer (see [4]) in this direction. They define an invariant which they call analytic torsion. The Ray-Singer analytic torsion is