Riemaniann metric problem in K&N's book.











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
    Nov 18 at 22:57










  • You are right but it's a classic book.
    – Hurjui Ionut
    Nov 19 at 8:23















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
    Nov 18 at 22:57










  • You are right but it's a classic book.
    – Hurjui Ionut
    Nov 19 at 8:23













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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 19 at 8:24

























asked Nov 18 at 16:51









Hurjui Ionut

456211




456211












  • Do you think everbody knows who K&N is?
    – Paul Frost
    Nov 18 at 22:57










  • You are right but it's a classic book.
    – Hurjui Ionut
    Nov 19 at 8:23


















  • Do you think everbody knows who K&N is?
    – Paul Frost
    Nov 18 at 22:57










  • You are right but it's a classic book.
    – Hurjui Ionut
    Nov 19 at 8:23
















Do you think everbody knows who K&N is?
– Paul Frost
Nov 18 at 22:57




Do you think everbody knows who K&N is?
– Paul Frost
Nov 18 at 22:57












You are right but it's a classic book.
– Hurjui Ionut
Nov 19 at 8:23




You are right but it's a classic book.
– Hurjui Ionut
Nov 19 at 8:23















active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003778%2friemaniann-metric-problem-in-kns-book%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003778%2friemaniann-metric-problem-in-kns-book%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Aardman Animations

Are they similar matrix