A Riemannian metric on a smooth manifold \(M\) is a family of inner products \[g_p : T_pM \times T_pM \;\longrightarrow\; \mathbb{R}, \qquad p \in M,\] varying smoothly in \(p\), such that each \(g_p\) is symmetric and positive-definite. In local coordinates the metric is completely determined by its values on basis tangent vectors: \[g_{ij}(p) \;:=\; g_p\!\left(\frac{\partial}{\partial x^i}\bigg|_p,\; \frac{\partial}{\partial x^j}\bigg|_p\right), \qquad g_{ij} = g_{ji},\] with the matrix \((g_{ij}(p))\) positive-definite at every point. The length of a tangent vector \(v = \sum_i v^i \frac{\partial}{\partial x^i}\in T_pM\) is then \(\|v\|_g = \sqrt{\sum_{i,j} g_{ij}(p)\, v^i v^j}\).
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My favourite thing to see on the feeds, though, is the evidence that there are humans still out there generating content. Sometimes an article will have a temporary headline with "TKTKTK" in it, which was captured by the aggregator before it was updated to its final version. The multiple outcome pieces especially make me chuckle: I, too, have been a web editor under pressure to get an article out the second the polls close or a casting is announced, making multiple versions on the sly in the CMS and then only promoting the one that turned out to be correct to the homepage.
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有开发者构建的 OpenClaw 实例扫描器，观测数据显示目前暴露在公网上的实例合计共有 23.49 万个，其中少部分实例已经是离线状态，大部分实例还是在线状态。，更多细节参见体育直播
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