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Induced metric

Let U be an open set in ℝ, and let φ be a continuously differentiable function from U into the Euclidean space ℝ, where m > n. The mapping φ is called an immersion if its differential is injective at every point of U. The image of φ is called an immersed submanifold. More specifically, for m = 3, which means that the ambient Euclidean space is ℝ, the induced metric tensor is called the first fundamental form.

Suppose that φ is an immersion onto the submanifold M ⊂ R. The usual Euclidean dot product in ℝ is a metric which, when restricted to vectors tangent to M, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.

Suppose that v is a tangent vector at a point of U, say

where e are the standard coordinate vectors in ℝ. When φ is applied to U, the vector v goes over to the vector tangent to M given by

(This is called the pushforward of v along φ.) Given two such vectors, v and w, the induced metric is defined by

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields e is given by

where Dφ is the Jacobian matrix: