A manifold is a topological space which locally looks like R^n. Calculus on a manifold is assured by the existence of smooth coordinate system.(adsbygoogle = window.adsbygoogle || []).push({});

A manifold may carry afurtherstructure if it is endowed with a metric tensor.

Why further structure?

If have sphere or a cylinder I can parameterize it with different coordinates ( two patches for sphere), which means different metrics. But does not have any manifold a canonical metric from the start? What is meant with endowing a manifold with a metric?

If I got a cylinder, its metric is euclidean by definition, no need for defining a extra structure on it.

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# Riemannian manifolds

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