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Buku Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski & Paweł Walczak

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski & Paweł Walczak

Author:Vladimir Rovenski & Paweł Walczak

Language: eng

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Publisher: Springer New York, New York, NY

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski & Paweł Walczak

(3.8)

For any tensor h(b) of the form , see (3.1), we have

(3.9)

 

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski & Paweł Walczak

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski & Paweł Walczak

Author:Vladimir Rovenski & Paweł Walczak , Date: September 29, 2019

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Author:Vladimir Rovenski & Paweł Walczak

Language: eng

Format: epub

Publisher: Springer New York, New York, NY
(3.8)

For any tensor h(b) of the form , see (3.1), we have

(3.9)

Proof.

This follows from the following computation (see also Lemma 2.3).

Notice that for any vector , the vector ϕ ∗  X = dϕ(X) is tangent to . From the definition of the metric g 1, we have g 1(X, Y ) = σ ⋅g 2(ϕ ∗  X, ϕ ∗  Y ) for . Let N 2 be a unit normal of . Then N 1 = ϕ ∗   − 1 N 2 is a unit normal of . In fact,

For the second fundamental form of , we obtain a symmetric (0, 2)-tensor field . Denote by ∇ (i) the Levi-Civita connection of (M i , g i ). For arbitrary vector fields we have

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