Who thinks it's simply connected?
谁认为它是单连通的？
Who thinks it's not simply connected?
谁认为它不是单连通的？
Or, as we will see soon, in a simply connected region.
稍后我们会知道，在一个单连通区域上也可以。
And, that's where the assumption of simply connected is useful.
在这里单连通的假设是有用的。
OK, so anyway, that's the customary warning about simply connected things.
总之，这是关于单连通的习惯性提醒。
OK, so space with the origin removed, OK, you think it's simply connected?
全空间去掉原点，它是单连通的吗？
OK, so that's the definition of the surface of a sphere being simply connected.
这就是球面上，对单连通的定义。
So, that's one way to check just for sure that this thing is not simply connected.
这是检验,该区域不是单连通的一种方法。
OK, so where do we use the assumption of being defined in a simply connected region?
那么我们是在哪里利用了，“定义在单连通区域”的假设呢？
Any object with a simply connected surface can be smoothed out to look like a sphere.
任何具有单连通表面的物体都能简化成一个球体。
In many ways, I simply connected the dots in these different strands of thinking and warnings.
在许多方面，我仅仅只是把许多不同思考和警示的线索联系起来。
As mathematicians say, “Every simply connected closed 3-manifold is homeomorphic to a 3-sphere.”
用数学语言表达就是“每个单连通3重（3-manifold）封闭体与一个三维球体同胚（homeomorphic）。”
As mathematicians say, "Every simply connected closed 3-manifold is homeomorphic to a 3-sphere."
用数学语言表达就是“每个单连通3重（3-manifold）封闭体与一个三维球体同胚（homeomorphic）。”
I mean, we will shed more light on this a bit later with the notion of simply connected regions and so on.
稍后我们将要进一步，讲解这一概念的相关知识。
Let's not worry too much about it. For accuracy we need our vector field to be defined in a simply connected region.
对于它，不用太担心，为了精确起见，需要向量场定义在一个单连通区域中。
Then, a new kind of so-called stratified triangulation of a simply connected planar polygonal region is introduced.
文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数。
Spacetime also appears to have a simply connected topology, at least on the length-scale of the observable universe.
时空似乎也有一个简单连接的拓扑结构，至少的长度，规模之大可见的宇宙。
In this example, the surface is not simply connected and any smoothed-out object looks like a torus with at least one hole.
在这个例子里，圈饼表面就是非单连通的，并且这类物体的简化就会像是至少有一个洞的曲面。
OK, so for this guy, domain of definition, which is plane minus the origin with the origin removed is not simply connected.
好滴，目前这里，在有定义的区域里，减掉原点，即移去原点，就不是单连通的。
Also, the paper discuss the existence of the infinite closed geodesics of a compact no-simply connected Riemannian manifold.
并由此讨论了紧致的非单连通黎曼流形上无穷多的闭测地线存在性问题。
With the help of them, it can be proved that the non-simply connected compact surface with nonnegative curvature must be flat.
讨论了紧致非单连通的具非负曲率的流形的一些几何性质，并应用它们证明了具非负曲率的紧致非单连通曲面必为平坦的。
This concept of being simply connected or not, and studying which loops bound surfaces or not can be used to classify shapes of things inside space.
单连通和非单连通的概念，研究哪些环路能界定曲面，可以用来对空间内部物体的形状进行分类。
With the help of the conformal transformation, we transform a simply connected region and its boundary into the upper half plane and real axis.
借助保形变换将单连通区域及其边界化为上半平面与实轴。
And here, to be completely truthful, I have to say defined in a simply connected region. Otherwise, we might have the same kind of strange things happening as before.
这里，为了使得完全成立，不得不假设，这定义在一个单连通区域，否则就有可能得到，和先前一样的奇怪事情。
An algorithm is presented for converting the linear quadtree representation of a simply connected region into a 4-direction chain code description of the region's boundary.
本文提出一种算法实现单连通区域的线性四元树表示转换成区域边界的4 -方向链码描述。
OK, so, we've seen that if we have a vector field defined in a simply connected region, and its curl is zero, then it's a gradient field, and the line integral is path independent.
一个向量场,如果定义在单连通区域并且旋度为零，那么它就是一个梯度场，并且其上的线积分与路径无关。
If these conditions are satisfied, the simply connected spring-mass system may be constructed uniquely. An algorithm is used to construct the physical realizable system from the data.
如果这些条件得到满足，则可惟一地构造简单连接弹簧-质点系统，并给出了构造真实物理系统的一个算法。
So, if we really wanted to apply Stokes theorem, because I've been telling you that space is simply connected, and I will always be able to apply Stokes theorem to any curve, what would I do?
因为我已经告诉过你们空间是单连通的，所以，如果我们真的想要运用Stokes定理，那么我总是可以把它运用于任何曲线，我应该做什么呢？
Well, if you know that your vector field is defined everywhere in a simply connected region, then you don't have to worry about this question of, can I apply Green's theorem to the inside?
如果知道了，向量空间在单连通区域内处处有定义，那么就可以毫无顾忌地，在这个区域里使用格林公式了？
And, maybe on the final it would be, there won't be any really, really complicated things probably, but you might need to be at least vaguely aware of this issue of things being simply connected.
期末考也可能会用得上，实际上，真的不会有十分复杂的问题，不过你们也要大概地了解一下,这个关于单连通区域的知识点。
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