Symmetric Spaces
Symmetric Spaces
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DOI：
10.1090/gsm/034/04
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发表时间：
2010
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通讯作者：
Andrew Fiori
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文献类型：
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作者：
Andrew Fiori
·对称空间的一个特性是，每个点都有一个全局对称性，通过该点“反转“测地线。定义.一个黎曼流形M称为局部对称的，如果对于每个点p ∈ M，存在p的正规邻域U使得映射j p = exp p ·（−Id）· exp ·映射j p具有“反转“通过点p的测地线的性质。这意味着如果γ v：（−ε，ε）→ U <$M是（唯一的）几何，其中γ v（0）= p且γ v（o）= v ∈ T p M，则j p（γ v（t））= γ v（−t）。实际上，由于γ v（t）= exp p（tv），我们得到jp（γ v（t））= exp ·（−Id）（tv）= exp p（−tv）= γ v（−t）。因此，映射j p称为局部测地线对称或简称为局部对称。·很明显，j 2 P =Id，如果v ∈ T p M，则（dj p）p（v）=（dj p）p（γ v（0））=（j p · γ v）（0）= γ −v（0）= −v因此（dj p）p = −Id p。Cartan定理具有曲率张量R的黎曼流形是局部对称空间当且仅当DR = 0。（10）注意，我们已经有了映射的条件;即，如果ε很小，使得exp p：B ε（0 p）→ B ε（p）是一个单同态，那么我们可以在这些坐标中定义j p（x）= −x。当我们有平行曲率张量时，为什么这是等距的，这还有待观察。- 要看到这一点，我们必须证明在这些坐标中，度规在x和−x处是相同的。
• A characteristic property of a symmetric space is that every point has a global symmetry that " reverses " the geodesics through that point. Definition. A Riemannian manifold M is called locally symmetric if for every point p ∈ M there exists a normal neighborhood U of p such that the map j p = exp p • (−Id) • exp • The map j p has the property " reversing " the geodesics that pass through the point p. This means that if γ v : (−ε, ε) → U ⊂ M is the (unique) geometric with γ v (0) = p and γ v (o) = v ∈ T p M, then j p (γ v (t)) = γ v (−t). Indeed, since γ v (t) = exp p (tv), we obtain that j p (γ v (t)) = exp • (−Id)(tv) = exp p (−tv) = γ v (−t). • For this reason the map j p is called a locally geodesic symmetry or simply a local symmetry. • Forthermore, it is obvious that j 2 P =Id, and if v ∈ T p M , then (dj p) p (v) = (dj p) p (γ v (0)) = (j p • γ v) (0) = γ −v (0) = −v hence (dj p) p = −Id p. Such an isometry is called an involution. Cartan's Theorem. A Riemannian manifold with curvature tensor R is a locally symmetric space iff DR = 0. (⇐) Note that we already have a conditate for a map; namely, if ε is so small that exp p : B ε (0 p) → B ε (p) is a diffeomorphism, then we can just define j p (x) = −x in these coordinates. It remains to see why this is an isometry when we have parallel curvature tensor. – To see this, we must show that in these coordinates the metric is the same at x and −x.