<ruby id="d9npn"></ruby>

      <sub id="d9npn"><progress id="d9npn"></progress></sub>

      <nobr id="d9npn"></nobr>

      <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

      <th id="d9npn"><meter id="d9npn"></meter></th>

      Stack Exchange Network

      Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

      Visit Stack Exchange

      Welcome to MathOverflow

      MathOverflow is a question and answer site for professional mathematicians. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about research level mathematics.

      We're a little bit different from other sites. Here's how:


      Ask questions, get answers, no distractions

      This site is all about getting answers. It's not a discussion forum. There's no chit-chat.

      Just questions...

      ...and answers.

      up vote

      Good answers are voted up and rise to the top.

      The best answers show up first so that they are always easy to find.

      accept

      The person who asked can mark one answer as "accepted".

      Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.

      A non integrable distribution which is totally geodesic

      up vote 14 down vote favorite

      Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense:

      Every geodesic whose tangent vector of its intitial point is tangent to the distribution then the tangent vector at all its points is tangent to $D$, too.

      2 Answers

      up vote 4 down vote accept

      Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-planes.

      up vote 3 down vote

      Take $\mathbb{R}^3$ with the distribution which is the kernel of the one-form $dz - y dx$. This is the standard example of a contact structure on $\mathbb{R}^3$. See https://en.wikipedia.org/wiki/Contact_geometry .


      Get answers to practical, detailed questions

      Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.

      Ask about...

      • Specific issues with research level mathematics
      • Real problems or questions that you’ve encountered

      Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.

      Questions that need improvement may be closed until someone fixes them.

      Don't ask about...

      • Anything not directly related to research level mathematics
      • Questions that are primarily opinion-based
      • Questions with too many possible answers or that would require an extremely long answer

      Tags make it easy to find interesting questions

      All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.

      Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.

      A non integrable distribution which is totally geodesic

      up vote 14 down vote

      Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense:

      Every geodesic whose tangent vector of its intitial point is tangent to the distribution then the tangent vector at all its points is tangent to $D$, too.


      You earn reputation when people vote on your posts

      Your reputation score goes up when others vote up your questions, answers and edits.

      +5 question voted up
      +10 answer voted up
      +15 answer is accepted
      +2 edit approved

      As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.

      Reputation Privilege
      15 Vote up
      50 Leave comments
      125 Vote down (costs 1 rep on answers)

      At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.

      2000 Edit other people's posts
      3000 Vote to close, reopen, or migrate questions
      10000 Access to moderation tools
      see all privileges

      Improve posts by editing or commenting

      Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.

      Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.

      Use comments to ask for more information or clarify a question or answer.

      You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.

      Remember: we're all here to learn, so be friendly and helpful!

      up vote 9 down vote

      Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-planes.

      edit

      @AliTaghavi. it is a good question. Some years ago Patrick Massot, a student of Giroux, did some nice work on this: projecteuclid.org/euclid.gt/1513800108 - alvarezpaiva Jul 26 '18 at 14:37

      add a comment


      Unlock badges for special achievements

      Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.

      In fact, you can earn a badge just for reading this page:

       Informed Read the entire tour page
       Student First question with score of 1 or more
       Editor First edit
       Good Answer Answer score of 25 or more
       Civic Duty Vote 300 or more times
       Famous Question Question with 10,000 views

      see all badges


      Sign up to get started

      Signing up allows you to:

      • Earn reputation when you help others with questions, answers and edits.
      • Select favorite tags to customize your home page.
      • Claim your first badge:  Informed
      Looking for more in-depth information on the site? Visit the Help Center

      MathOverflow is part of the Stack Exchange network

      Like this site? Stack Exchange is a network of 174 Q&A sites just like it. Check out the full list of sites.

      Stack Exchange

      特码生肖图
      <ruby id="d9npn"></ruby>

          <sub id="d9npn"><progress id="d9npn"></progress></sub>

          <nobr id="d9npn"></nobr>

          <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

          <th id="d9npn"><meter id="d9npn"></meter></th>

          <ruby id="d9npn"></ruby>

              <sub id="d9npn"><progress id="d9npn"></progress></sub>

              <nobr id="d9npn"></nobr>

              <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

              <th id="d9npn"><meter id="d9npn"></meter></th>