definition of injective immersion
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3 Immersions and Embeddings
Definition: (Embeddings) An immersion ϕ : M → N of differentiable mani- folds is an embedding if ϕ is a homeomorphism of M onto its image ϕ(M) ⊂ N where ϕ(M) |
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Lecture 9: the whitney embedding theorem
Behind Step 3: Injective immersions are locally nice compactness means there are only finitely many local pieces 1 The Whitney embedding theorem: Compact |
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CHAPTER 6 IMMERSIONS AND EMBEDDINGS In this
Definition 6 1*** Suppose f : N → M is a smooth map between manifolds The map f is called an immersion if f∗x : TxN → Tf(x)M is injective for all x ∈ N |
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Embedding and immersion theorems
Zoe.pdf |
What is the difference between immersion and embedding?
Differential topology
is called an immersion if its derivative is everywhere injective.
An embedding, or a smooth embedding, is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).Consider the circle S1 = {x ∈ R2 : x = 1}.
Now consider the function f : S1 → R where f(θ) = θ2 with θ(x) being a local coordinate system on S1.
Then the derivative df(θ)=2θ, which is injective.
So f is an immersion.
Is an immersion injective?
The map f is called an immersion if f∗x : TxN → Tf(x)M is injective for all x ∈ N.
The derivative is injective at each point is not enough to guarantee that the func- tion is one-to-one, as very simple example illustrate.
Take f : R → R2 by f(x) = (sin(2πx), cos(2πx)).
What is an immersion in geometry?
In differential geometry, an immersion is a function that maps one manifold into another.
Immersions are only possible if the manifolds have certain properties.
The Klein Bottle immersed in three-dimensional space.
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EMBEDDING AND IMMERSION THEOREMS Contents 1
Zoe.pdf |
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THE WHITNEY EMBEDDING THEOREM As we mentioned in
the extrinsic/concrete definition used by Poincaré (as the set of possible Behind Step 3: Injective immersions are locally nice compactness means there. |
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Expressing an observer in preferred coordinates by transforming an
19 févr. 2018 trajectories of (x ˆw) may leave the domain of definition of the ... given by the injective immersion ?? defined in (1.3) leads to the ... |
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Expressing an observer in preferred coordinates by transforming an
19 févr. 2018 In the example above pulling the observer dynamics in the ... given by the injective immersion ?? defined in (1.3) leads to the function ... |
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3 Immersions and Embeddings
Definition (Immersions) A differentiable mapping. ? : Mm ? Nn of differentiable manifolds is said to be an immersion if d?p : TpM ? T?(p)N is injective |
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CHAPTER 6 IMMERSIONS AND EMBEDDINGS In this chapter we
Definition 6.1***. Suppose f : N ? M is a smooth map between manifolds. The map f is called an immersion if f?x : TxN ? Tf(x)M is injective for all x |
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Untitled
2 août 2018 The map f is called an embedding if it is a proper injective immersion. Remark 2.3.3. In our definition of proper maps it is important that the. |
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SMOOTH SUBMANIFOLDS 1. Smooth submanifolds Let M be a
Roughly speaking smooth submanifolds are objects that are defined locally by equations In general |
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1 September 12 2014
26 nov. 2014 Definition 5.5. An immersion f : X ? Y is an embedding if f is injective and proper. 6 September 24 2014. |
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LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically
In 1912 Weyl gave an intrinsic definition for smooth manifolds. Proof. Suppose we already have an injective immersion ? : M ? RK with K > 2m+1. |
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EMBEDDING AND IMMERSION THEOREMS Contents 1
an injective immersion defined over it is automatically proper, so is an embedding Definition 2 7 The tangent bundle of Mk ⊂ Rn is denoted TMk where |
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3 Immersions and Embeddings - UCSD Math
Definition (Immersions) A differentiable mapping ϕ : Mm → Nn of differentiable manifolds is said to be an immersion if dϕp : TpM → Tϕ(p)N is injective for all |
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Math 205C - Topology Midterm Erin Pearse 1 a) State the definition
b) immersion An immersion is a differentiable mapping f : M → N such that dim M = rankf at every point of M d) imbedding An imbedding is an injective immersion f : M → N which is homeomorphic to its image f(M) ⊂ N |
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LECTURE 8: SMOOTH SUBMANIFOLDS 1 Smooth submanifolds
Example The following two graphs are the images of two immersions of R into R2 For the first one, the immersion is not injective For the |
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MANIFOLDS MA3H5 PART 2 4 Immersions and submersions We
Any proper injective immersion is an embedding Proof This is an immediate consequence of the fact (mentioned in Section 1, that a continuous injective map |
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Notes
Example 1 1 3 The three conditions defining a topological manifold are independent example, we have an injective immersion which is not proper Definition |
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11 Submanifolds
an open subset U of Rk and an immersion ϕ : U → Rn such that ϕ : U → ϕ(U) is if Tpf is injective (surjective) for every p ∈ M If dim(M) = m and dim(N) = n Proof By the above, the rank of f is independent of the chosen charts, so without |
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Here
2 août 2018 · an embedding if it is a proper injective immersion sections 5,6,7 Remark 2 3 3 In our definition of proper maps it is important that the compact |
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Chapter 5 Basic Differential Topology
Definition 5 18 A smooth injective immersion f : M ---+ N between smooth manifolds M and N is called an embedding if and only iff is a homeomor- phism onto its |
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Whitneys immersion and embedding theorems
Then the class of Cr injective immer- sions is dense in the space of Cr maps from Mm to Rs, with respect to the Whitney C1 topology This means: given f ∈ Cr(M; |
