Abstract
In this chapter, we introduce the reader to the theory of manifolds. We start with the very notion of a manifold, illustrate it by a number of examples and discuss level sets in some detail. Thereafter, we carry over the concepts of differentiable mapping, tangent space and derivative from classical calculus to manifolds and derive manifold versions of the Inverse Mapping Theorem, the Implicit Mapping Theorem and the Constant Rank Theorem. Next, we discuss submanifolds in some detail. We define them to be injective immersions and distinguish the important special classes of embedded and initial (or weakly embedded) submanifolds. Besides that, we derive criteria for a subset to admit a submanifold structure. Finally, we prove the Transversal Mapping Theorem, which states that the preimage of a submanifold under a differentiable mapping is again a submanifold, provided the mapping is transversal to that submanifold.
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Notes
- 1.
The coarsest topology such that both κ + and κ − are continuous.
- 2.
The coarsest topology such that the natural inclusion mapping Sn→ℝn+1 is continuous.
- 3.
Usually, we will denote elements of an algebra by capital A,B,C,… and elements of a group by small a,b,c,….
- 4.
We adopt the convention that scalars multiply from the right; this is of course relevant for \(\mathbb{K}= \mathbb{H}\) only.
- 5.
In this book, mappings between manifolds are usually denoted by capital Greek letters like Φ,Ψ,… or small Greek letters like φ,ψ,χ,….
- 6.
If there is no danger of confusion, we will usually omit the chart label and just write \(X_{m}^{i}\).
- 7.
As for the local representative of a tangent vector, we will usually omit the chart label and just write ∂ i,m .
- 8.
All such measures are equivalent, that is, they have the same sets of measure zero.
- 9.
The image of this curve is not a figure eight, because it is missing the point (0,1).
- 10.
By definition, this means that the induced mapping \(\tilde{\varphi}\) is open with respect to the relative topology. In this case, φ is called an embedding.
- 11.
Or, alternatively, weakly embedded.
- 12.
In fact, they coincide with the connected components of A.
- 13.
Since M already carries a topology, there is no need to require the family {(V i ,ρ i )} to be countable.
- 14.
In other words, φ is a subimmersion at every m 0∈M, cf. Exercise 1.5.5/(a).
- 15.
Since the connected components of M may have different dimensions, M as a whole may not be a manifold. We leave it to the reader to provide examples.
- 16.
That is, the codimension of M in N equals the codimension of Q in P.
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Rudolph, G., Schmidt, M. (2013). Differentiable Manifolds. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_1
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