Locally Conformally Flat Manifolds And Weyl And Cotton Tensors


The purpose of this note is to prove the following
Theorem. A Riemannian manifold (M^n, g) is locally conformally flat if and only if
  • for n \geqslant 4, the Weyl tensor vanishes;
  • for n=3, the Cotton tensor vanishes.
To this purpose, let us briefly recall some definitions
The Weyl tensor. The Weyl tensor can be defined using the following formula
\displaystyle W = \text{Rm} - \frac{\text{Scal}}{{2(n - 1)n}}g \odot g - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{\text{Scal}}{n}g} \right) \odot g
where n\geqslant 3 and \odot denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors. Writing the Weyl tensor in this way means that the Weyl tensor is actually a (0,4) tensor. It can be seen that the Weyl tensor can be rewritten in this form
\displaystyle W = \text{Rm} - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) \odot g
where the part
\displaystyle S = \frac{1}{{n - 2}}\left( {{\text{Ric}} - \frac{g}{{2(n - 2)}}{\text{Scal}}} \right) \odot g
is called the Schouten tensor. We have the following result
Proposition. If n \geqslant 3, then
\displaystyle {\nabla ^l}{W_{lijk}} = \frac{{n - 3}}{{n - 2}}{C_{ijk}}
where
\displaystyle {C_{ijk}} = {\nabla _k}{S_{ij}} - {\nabla _i}{S_{kj}}.
The Cotton tensor. The (0,3) tensor C above is called the Cotton tensor. Apparently, if either the Weyl tensor or the Ricci tensor vanishes, so does the Cotton tensor.
The Weyl and Cotton tensors under the conformal changes of metric. It is well-known that these tensors are invariant under the conformal changes of metric, that is,
\displaystyle W = \widetilde W, \quad C = \widetilde C
under the change \widetilde g = e^{2f}g for some smooth function f (see this note).
The Riemmanian curvature tensor under the conformal changes of metric. We list here the following rule
\displaystyle {e^{ - 2f}}\widetilde {\text{Rm}} = \text{Rm} - \left( {{\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}}} \right) \odot g.
See this note for further details.
Locally conformally flat manifolds. Roughly speaking, this is to say at each point p \in M, there exists a neighborhood U of p such that the conformal class of g contains the flat metric in U, that is to say
\displaystyle \widetilde{\text{Rm}} =0.
We are now in a position to prove the theorem.
Proof of Theorem. We first assume that M is locally conformally flat, that is, \widetilde{\text{Rm}} =0. If n\geqslant 4, using the formula for W we have
\displaystyle W = \widetilde W = \widetilde {\text{Rm}} - \frac{{\text{Scal}_{\widetilde g}}}{{2(n - 1)n}}\widetilde g \odot \widetilde g - \frac{1}{{n - 2}}\left( {\widetilde {\text{Ric}} - \frac{{\text{Scal}_{\widetilde g}}}{n}\widetilde g} \right) \odot \widetilde g = 0
since the Riemmanian curvature tensor vanishes. If n=3, we use the formula for C, we obtain
\displaystyle {C_{ijk}} = {\widetilde C_{ijk}} = {\widetilde {{\text{Ric}}}_{ij,k}} - {\widetilde {{\text{Ric}}}_{ik,j}} = 0
since the Ricci tensor vanishes.
Conversely, if the Weyl tensor vanishes, we have
\displaystyle 0 = \text{Rm} - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) \odot g.
Under the conformal change, for some f, we have
\displaystyle {e^{ - 2f}}\widetilde {Rm} = \left[ {\frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) - \left( {{\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}}} \right)} \right] \odot g.
Since the mapping \odot: S^2 M \to CM given by \odot (h) = h \odot g is injective, it suffices to show that the following equation
\displaystyle \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) = \left( {{\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}}} \right).
is locally solvable. This can be done using the following whose proof is postponed.
Lemma. Provided the Weyl tensor vanishes, equation
\displaystyle {\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}} = {S_{ij}}
is locally solvable if and only if the following integrability condition is satis ed
\displaystyle {\nabla _k}{S_{ij}} = {\nabla _i}{S_{kj}}.
That is, if and only if the Cotton tensor vanishes.
Recall that when n\geqslant 4; the condition follows from the Weyl tensor vanishes. This concludes the proof.

6 comments:

  1. Wash your curly hair at the very least two days in a week. This article
    will explain the hair product diversion conspiracy and list
    ways that you can determine whether or not the beloved salon brand you saw in Walgreens last week was truly diverted merchandise or strategically placed by the powers that
    be. Soon after this, use curl lotion all more than your hair, and make use of mousse and squeeze
    this in your hair to that your curls seem flawless.

    Also visit my web blog - hair products

    ReplyDelete
  2. Nanokeratin locks onto the hair, forming a fine,
    smooth coat of keratin. As natural products don't have any side-effects, you can rest assured for healthy hair. The product should be one that is made for your type of hair, whether it be dry, oily, curly or straight.

    My weblog :: hair products

    ReplyDelete
  3. 37 kits (round that up to 9 total quart and a half kits).
    First, ready mixed concrete in Sunderland can help you get your construction job done faster.
    It is advantageous for the customer to compare the prices for an additional dumpster rental from the normal
    waste removal company against an offer made by a collector
    that will place a dumpster for a one-time pickup.

    ReplyDelete
  4. Use oils that can help keep your hair soft and manageable.
    - If your hair feels rough, condition it right away. You repeat the exact same procedure maybe once
    or twice every week. The remaining oil in the scalp for a day and
    then follow up with a shampoo the next morning.

    Customers have always given Moroccan Oil products the highest ratings.


    Feel free to visit my blog post - how to grow hair faster

    ReplyDelete
  5. It is indeed possible to lower high blood pressure by only taking herbal medicines and vitamins and eating certain foods that can lower high blood pressure.

    It needs to be rushed to a medical facility which is capable
    of handling emergencies. It is also possible that if CLASS recipients are eventually enrolled in AHCCCS (Arizona's Medicaid program), AHCCCS will benefit financially by receiving a share of the cash payment made by the CLASS program. , one of the chiropractors in Fayetteville, is pleased to announce the implementation of programs for natural detoxification therapy. Finally, I think anyone would be inspired by Lynda's story.


    Feel free to surf to my web blog ... mixobarbaric

    ReplyDelete
  6. Aside from Microsoft and Nintendo, another top competitor, Sony,
    will also be releasing some top-rated games for its own console - the Playstation.
    It is also a good practice to make sure that none of the climbers are more tired than the other.
    However, such training programs are tough to wave through since it require tons
    of learning patience and willingness to proven one’s innate skills.
    The main purpose of a simulator is to understand the behavior of the
    system. The instruments that lag in real
    life, lag correctly, gyro drift is modeled correctly, the magnetic
    compass is subject to aircraft body forces - all those things that make
    real world flying a challenge are present.

    ReplyDelete