bending of manifold and coordinate change

I am new to this subject of topology. I want to know if bending and stretching of a manifold is same as a general transformation of a coordinate system drawn on the manifold. Or the mathematical definition of bending and stretching shall equally help.
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 Recognitions: Gold Member Homework Help Science Advisor Deforming a local coordinate system of a manifold, you only get a new coordinate system for the same manifold. Mathematically, I would describe the bending or stretching of a manifold M embedded in some R^n as an homotopy h:[0,1] x M --> R^n which is an embedding at all times. Given a metric on the manifold, there may be a more intrinsic description of bending\stretching.
 There is then another result you may be able to use here: Say M is your manifold in its initial state, and M' is the "final" state of the manifold, after doing some bending and stretching. If the bending and stretching are done homeomorphically, i.e., if there is a homeo. h between M and M' , then you can use h to pullback the charts of M into those of M'. If the bending and stretching are not a homeomorphisms, i.e., if the homotopy that Quasar described is not an isotopy, then you are dealing with a manifold M' that is topologically different from M, i.e., there may not be atlases for M' that are compatible with those for M.

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bending of manifold and coordinate change

 Quote by 01030312 I am new to this subject of topology. I want to know if bending and stretching of a manifold is same as a general transformation of a coordinate system drawn on the manifold. Or the mathematical definition of bending and stretching shall equally help.
coordinate transformations have nothing to do with the shape of the manifold. Stretching is a change of shape and must be represented as a change of metric not as a coordinate transformation.

Bending has meaning only for an embedded manifold. Here the bending does not change the metric but rather changes the embedded curvature - which is an extrinsic geometrical property of the embedding. Again this has nothing to do with coordinate transformations.
 "quasar" and "lavinia" Here is my real problem. Consider a n-plane. And take two parallel lines on it. Any deformation of this plane will not affect the relation between lines- that they never meet. The riemann curvature tensor (or some tensors of raychaudhari equation), which governs this relation and is zero in n-plane, should vanish even in deformed n-surface. So vanishing of tensors can be obtained by any coordinate transform of the coordinate on plane (except the case of bending). Unless there is some other transform too, after which the tensors vanish, I am having quite a difficulty in understanding that deformation is not a coordinate transform.
 "lavinia" Given a new metric g'ab and an old one gab, both infinitesimally different, we can define lie derivative of a vector field 'Zi' with respect to the metric g to be (g'-g)/e , where e ->0, and transform coordinates by xi -> xi + eZi , e->0. With this transform we can get the original metric, for deformed manifold, thus identifying stretching again with coordinate change. ( I am not sure if i am correct or not. Expert's comments required).

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