Riemaniann metric problem in K&N's book.











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I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170:



enter image description here



And the proof gose like this:



enter image description here



My problem is that I don't understand why we can assume the last double inequality. My only idea is to look at $g$ in normal coordinates and thus $g_{ij}(x)=delta_{ij}-frac{1}{3}R_{iajb}x^ax^b+O(epsilon^3).$ Can someone help me with some details please?










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  • Do you think everbody knows who K&N is?
    – Paul Frost
    10 hours ago










  • You are right but it's a classic book.
    – Hurjui Ionut
    1 hour ago















up vote
3
down vote

favorite












I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170:



enter image description here



And the proof gose like this:



enter image description here



My problem is that I don't understand why we can assume the last double inequality. My only idea is to look at $g$ in normal coordinates and thus $g_{ij}(x)=delta_{ij}-frac{1}{3}R_{iajb}x^ax^b+O(epsilon^3).$ Can someone help me with some details please?










share|cite|improve this question
























  • Do you think everbody knows who K&N is?
    – Paul Frost
    10 hours ago










  • You are right but it's a classic book.
    – Hurjui Ionut
    1 hour ago













up vote
3
down vote

favorite









up vote
3
down vote

favorite











I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170:



enter image description here



And the proof gose like this:



enter image description here



My problem is that I don't understand why we can assume the last double inequality. My only idea is to look at $g$ in normal coordinates and thus $g_{ij}(x)=delta_{ij}-frac{1}{3}R_{iajb}x^ax^b+O(epsilon^3).$ Can someone help me with some details please?










share|cite|improve this question















I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170:



enter image description here



And the proof gose like this:



enter image description here



My problem is that I don't understand why we can assume the last double inequality. My only idea is to look at $g$ in normal coordinates and thus $g_{ij}(x)=delta_{ij}-frac{1}{3}R_{iajb}x^ax^b+O(epsilon^3).$ Can someone help me with some details please?







riemannian-geometry inner-product-space






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edited 1 hour ago

























asked 17 hours ago









Hurjui Ionut

451211




451211












  • Do you think everbody knows who K&N is?
    – Paul Frost
    10 hours ago










  • You are right but it's a classic book.
    – Hurjui Ionut
    1 hour ago


















  • Do you think everbody knows who K&N is?
    – Paul Frost
    10 hours ago










  • You are right but it's a classic book.
    – Hurjui Ionut
    1 hour ago
















Do you think everbody knows who K&N is?
– Paul Frost
10 hours ago




Do you think everbody knows who K&N is?
– Paul Frost
10 hours ago












You are right but it's a classic book.
– Hurjui Ionut
1 hour ago




You are right but it's a classic book.
– Hurjui Ionut
1 hour ago















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