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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MacBook Air with M5 was built with the environment in mind, and brings Apple even closer to reaching its ambitious plan to be carbon neutral across its entire footprint by 2030. It is made with 55 percent recycled content,8 including 100 percent recycled aluminum in the enclosure and 100 percent recycled cobalt in the battery. It is manufactured with 50 percent renewable electricity, such as wind and solar, across the supply chain. MacBook Air is designed to be durable and repairable, while meeting Apple’s high standards for energy efficiency and safer chemistry. The paper packaging is 100 percent fiber-based and can be easily recycled.9
US beverage and medical cannabis company Tilray has bought the company's UK brewing operations, brand and 11 pubs in a £33m deal.