how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?
i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.
i don't see how such an extention is done.
A smooth extension of a locally defined function on a manifold is a way of extending a function defined on a subset of a manifold to a larger domain while maintaining smoothness. This allows the function to be defined and continuous on the entire manifold, rather than just a portion of it.
Smooth extension is important in manifold theory because it allows for the study of functions on the entire manifold, rather than just on a subset. This can provide a more complete understanding of the manifold and its properties, and is often necessary in order to apply other mathematical tools and techniques.
A smooth extension of a locally defined function on a manifold can be achieved by using techniques such as partitions of unity or bump functions. These methods allow for the construction of a smooth function that agrees with the original function on the subset where it is defined, and is also smooth on the rest of the manifold.
In order for a smooth extension to exist, the original function must be continuous and at least C^k (k times continuously differentiable) on the subset of the manifold where it is defined. Additionally, the manifold must be a paracompact space, which ensures that a partition of unity exists.
No, not all locally defined functions on a manifold can be smoothly extended. There are certain functions that are not continuous or differentiable, and therefore cannot be extended in a smooth manner. Additionally, the existence of a smooth extension depends on the specific properties of the manifold and the function in question.