Let be an open set in , and let be a continuously differentiable function from into the Euclidean space , where . The mapping is called an immersion if its differential is injective at every point of . The image of is called an immersed submanifold.

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

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

where are the standard coordinate vectors in . When is applied to , the vector goes over to the vector tangent to given by

(This is called the pushforward of along .) Given two such vectors, and , 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 is given by

where is the Jacobian matrix:

স্থানাঙ্ক রূপান্তরের অধীনে পরিধির চাপের ধ্রুবক
Induced metric
Metric in a vector bundle
Signature of a metric