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Aug
14
revised How is a section viewed as a bundle map?
added 169 characters in body
Aug
14
comment How is a section viewed as a bundle map?
@Theo: Thanks! It is more clear now. I wonder if people think of a section more often as a "inverse" of projection, or as a bundle map? What is the purpose of viewing a section as a bundle map, which seems to me so indirect?
Aug
14
asked How is a section viewed as a bundle map?
Aug
14
comment Definition of manifold
@Hans: To make an analogy, in measure theory, a mapping from a measure space to a measurable space can induce a measure on the codomain.
Aug
14
accepted Understanding isomorphic equivalences of tensor product
Aug
14
comment Definition of manifold
@Hans: Thanks! Is there another purpose that the domain is endowed with the extra structure of the codomain via the inverse of the mapping?
Aug
14
accepted Meaning of “a mapping factors over another”?
Aug
14
comment Meaning of “a mapping factors over another”?
@gary: about the example in your broad description, for given f and g, can there be more than one mappings that make f factor over g?
Aug
14
asked Meaning of “a mapping factors over another”?
Aug
14
comment Definition of manifold
@Dylan: What does "trivial " mean?
Aug
14
comment Definition of manifold
Thanks! "when you define a manifold you DO NOT want to use any other structure of an euclidian space, just the topological structure." Then if replacing "Euclidean space" with a topological space in the definition of a topological manifold, will the definition still be the same? Similar question for the definition of a manifold, if replacing "a topological vector space over the reals" with a topological space.
Aug
14
comment Definition of manifold
@Ryan: That is because I am confused. "If you're going to say some object X is locally like object Y, how can you do that in a way that gives X more structure than Y?" is also my question.
Aug
14
revised Definition of manifold
added 59 characters in body
Aug
14
comment Definition of manifold
Thanks! How is the pull back like for a manifold and a topological manifold?
Aug
14
revised Definition of manifold
added 149 characters in body
Aug
14
revised Definition of manifold
added 15 characters in body
Aug
14
asked Definition of manifold
Aug
13
comment Understanding isomorphic equivalences of tensor product
@Zhen: The definition of tensor product I used is by a bilinear mapping with universal property.
Aug
13
asked Understanding isomorphic equivalences of tensor product
Aug
12
comment Tensors as matrices vs. Tensors as multi-linear maps
A third question: "Writing a (1,1)-tensor in this basis corresponds exactly to writing a linear transformation as a square matrix." When you write a matrix for a tensor, do you view the tensor as a linear transformation V→V, an element of $V^∗⊗V$, or a linear map $V⊗V^∗→R$?