<PAGE>

                                                                    EXHIBIT 10.6

                            NONCOMPETITION AGREEMENT


     NONCOMPETITION AGREEMENT (the "Agreement") dated as of this 27th day of
March, 2001, between New York Community Bancorp, Inc. a corporation organized
under the laws of Delaware, with its principal administrative office at 615
Merrick Avenue, Westbury, New York (the "Holding Company"); Richmond County
Savings Bank, a New York State chartered savings bank ("Richmond Bank");
Richmond County Financial Corp., a corporation organized under the laws of
Delaware and the principal holding company for Richmond Bank ("Richmond,"), both
Richmond Bank and Richmond having their principal administrative offices at 1214
Castleton Avenue, Staten Island, New York, and Anthony E. Burke (the
"Covenantor").

     WHEREAS, Covenantor is currently employed as President and Chief Operating
Officer of Richmond and President and Chief Operating Officer of Richmond Bank;

     WHEREAS, Richmond is entering into an Agreement and Plan of Merger of even
date herewith (the "Merger Agreement") with the Holding Company, pursuant to
which Richmond will merge with and into the Holding Company (the "Merger") with
the Holding Company being the surviving corporation and pursuant to which
Richmond Bank will merge with and into New York Community Bank, a New York State
chartered savings bank which is a wholly owned subsidiary of the Holding Company
(the "Bank"), with the Bank being the surviving bank; and

     WHEREAS, Covenantor and Richmond are parties to an employment agreement, as
amended and restated as of September 21, 1999, and Covenantor and Richmond Bank
are parties to an employment agreement dated as of September 21, 1999 (together,
the "Prior Employment Agreements");

     WHEREAS, Covenantor has entered into an Employment Agreement with the
Holding Company of even date herewith, pursuant to which Covenantor will serve
as Senior Executive Vice President and Chief Operating Officer of the Holding
Company and Covenantor has entered into an Employment Agreement with the Bank of
even date herewith pursuant to which Covenantor will serve as President and
Chief Operating Officer of the Bank (the "New Employment Agreements") effective
at the Effective Time as defined in the Merger Agreement (the "Effective Time");

     WHEREAS, in consideration of the willingness of the Holding Company and the
Bank to enter into the New Employment Agreements, Covenantor agrees that the
Prior Employment Agreements shall be cancelled as of the date hereof;

     WHEREAS, Covenantor agrees to enter into this Agreement in consideration
for which the Holding Company will provide certain monetary consideration
recited below;

     NOW, THEREFORE, in consideration of the foregoing and the mutual
commitments contained in this Agreement, the parties hereto agree as follows:
<PAGE>

     1.   Effectiveness; Effect on Prior Agreements.
          -----------------------------------------

          (a)  This Agreement shall be effective as of the date hereof. The
parties agree that the Prior Employment Agreements are null and void effective
as of the date hereof. Notwithstanding the foregoing, if the Merger Agreement is
terminated in accordance with Article VIII of the Merger Agreement, the Prior
Employment Agreements shall be deemed to have not been cancelled and this
Agreement shall be null and void.

     2.   Noncompete Consideration.
          ------------------------

     In consideration for his execution of this Agreement, the Holding Company
will pay to Covenantor the amount of $1,500,000 at the Effective Time provided
Covenantor is then employed by Richmond and Richmond Bank.

     3.   Confidential Information.
          ------------------------

          (a)  Covenantor agrees that he shall keep secret and confidential all
business-related information about Richmond, Richmond Bank, the Holding Company
and its subsidiaries, including without limitation, information about business
contacts, transactions, contracts, intellectual property, finances, personnel,
products and pricing, customers, or corporate affairs of which Covenantor may
have become aware, whether or not relating to or arising out of Covenantor's
specific duties, ("Confidential Information") and Covenantor shall not disclose
or make known any of such Confidential Information or anything relating thereto
to any person, firm or corporation except to officers, directors, employees,
agents and advisors of the Holding Company and its subsidiaries and such other
persons or entities as may be authorized by Richmond or the Holding Company;

          (b)  Upon termination of his employment with Richmond and Richmond
Bank or the Holding Company and its subsidiaries, Covenantor shall immediately
return to Richmond and Richmond Bank or the Holding Company and its subsidiaries
any and all Confidential Information in his possession or under his control,
including, without limitation, all reports, analyses, summaries, notes, or other
documents or work papers, containing or based upon any Confidential Information,
whether prepared by Richmond, Richmond Bank, the Holding Company or any of its
subsidiaries, Covenantor or any other person or entity.

          (c)  Should any person request in any manner that Covenantor disclose
any Confidential Information, Covenantor shall immediately notify Richmond and
Richmond Bank and the Holding Company and its subsidiaries of such request and
the content of all communications and discussions relating thereto.
<PAGE>

     4.   Covenant not to Compete; Nonsolicitation of Employees and Customers.
          -------------------------------------------------------------------

          (a)  Covenantor agrees that while employed by the Holding Company and
the Bank, and during the twelve month period following the termination of such
employment for any reason but in any event for a period of time no shorter than
the thirty-six month period immediately following the Effective Time, Covenantor
will:

          (i)   not, directly or indirectly (whether as principal, agent,
     independent contractor, employee or otherwise), own, manage, operate, join,
     control or otherwise carry on, participate in the ownership, management,
     operation or control of, or be engaged in or concerned with, any business
     competitive with that of the Holding Company or the Bank or their
     affiliates (a "Competing Business"), provided that the Covenantor shall not
     be prohibited from owning less than 5% of any publicly traded corporation,
     whether or not such corporation is in competition with the Holding Company
     or the Bank or their affiliates;

          (ii)  inform any person which seeks to engage the services of
     Covenantor that Covenantor is bound by this Section 4 and the other terms
     of this Agreement;

          (iii) not solicit or induce or attempt to solicit or induce, directly
     or indirectly, any employee of the Holding Company or the Bank or their
     affiliates, whether or not such person would commit a breach of any
     employment agreement by reason of leaving service, to terminate such
     employee's employment relationship with the Holding Company or the Bank or
     their affiliates in order to enter into any such relationship with him or
     any person in competition with the business of the Holding Company or the
     Bank or their affiliates;

          (iv)  not, (x) solicit by mail, by telephone, by personal meeting, or
     by any other means, either directly or indirectly, any customer or
     prospective customer of the Holding Company or the Bank or their affiliates
     to transact any Competing Business or to reduce or refrain from doing any
     business with the Holding Company or the Bank or their affiliates, or (y)
     interfere with or damage (or attempt to interfere with or damage) any
     relationship between the Holding Company or the Bank or their affiliates
     and any such customer or prospective customer.

          (b)   Covenantor understands and agrees that money damages may not be
a sufficient remedy for any breach or attempted or threatened breach of this
Section 4 by Covenantor and that the Holding Company and its subsidiaries shall
be entitled to equitable relief, including specific performance and injunctive
relief as remedies for any such breach or threatened or attempted breach.
Covenantor hereby consents to the granting of an injunction (temporary or
otherwise) against Covenantor or to the entering of any other court order
against Covenantor prohibiting and enjoining Covenantor from violating, or
directing Covenantor to comply with, any provision of this Section 4. Covenantor
also agrees and understands that such remedies shall be in addition to any
<PAGE>

and all remedies, including damages, available to the Holding Company or its
subsidiaries against Covenantor for such breaches or threatened or attempted
breaches.

          (c)  In addition to the Holding Company's rights set forth in
paragraph (b) of this Section 4, in the event that Covenantor shall violate the
terms and conditions of Sections 3 or 4 of this Agreement, the Holding Company
or the Bank may terminate any payments or benefits of any type and regardless of
source payable by the Holding Company or the Bank, if applicable, to Covenantor.

     5.   Entire Agreement; Modification.
          ------------------------------

     This Agreement contains the entire agreement between Covenantor, the
Holding Company, Richmond and Richmond Bank with respect to the subject matter
hereof and it is the complete, final and exclusive embodiment of our agreement
with regard to this subject matter hereof. It is entered into without reliance
on any promise or representation other than those expressly contained herein,
and it cannot be amended except in writing signed by all parties.

     In case any one or more of the provisions contained in this Agreement
shall, for any reason, be held to be invalid, illegal or unenforceable in any
respect, such invalidity, illegality or unenforceability shall not affect the
other provisions of this Agreement, and this Agreement shall be construed as if
such invalid, illegal or unenforceable provision had never been contained
herein. If, moreover, any one or more of the provisions contained in this
Agreement shall for any reason be held to be excessively broad as to duration,
geographical scope, activity or subject, it shall be construed by limiting and
reducing it, so as to be enforceable to the extent compatible with the
applicable law as it shall then appear.

     6.   Notices.
          -------

     For purposes of this Agreement, all notices and other communications
required or permitted hereunder shall be in writing and shall be deemed to have
been duly given when delivered or five (5) days after deposit in the United
States mail, certified and return receipt requested, postage prepaid, to such
address as either party may have furnished to the other in writing in accordance
herewith, except that notices of change of address shall be effective only upon
receipt.

     7.   Waiver.
          ------

     If either party should waive any breach of any provisions of this
Agreement, he or it shall not thereby be deemed to have waived any preceding or
succeeding breach of the same or any other provision of this Agreement.
<PAGE>

     8.   Assignment.
          ----------

     This Agreement and any rights or obligations hereunder may be assigned by
the Holding Company or the Bank to any successor in interest to the Holding
Company's or the Bank's business. This Agreement may not be assigned by
Covenantor.

     9.   Headings.
          --------

     The headings of the sections hereof are inserted for convenience only and
shall not be deemed to constitute a part hereof nor to affect the meaning
thereof.

     10.  Governing Law.
          -------------

     This Agreement shall be governed by the laws of the State of New York.

     11.  Required Provisions.
          -------------------

     Any payments made to Executive pursuant to this Agreement are subject to
and conditioned upon compliance with 12 U.S.C. ss.1828(k) and any rules and
regulations promulgated thereunder, including 12 C.F.R. Part 359.
<PAGE>

                                   SIGNATURES


         IN WITNESS WHEREOF, New York Community Bancorp, Inc., Richmond County
Financial Corp. and Richmond County Savings Bank have caused this Agreement to
be executed and its seal to be affixed hereunto by its duly authorized officer
and its directors, and Executive has signed this Agreement, on the date set
forth on the first page hereof.


ATTEST:                                      NEW YORK COMMUNITY BANCORP, INC.


/s/ Michael J. Lincks                        BY: /s/ Joseph R. Ficalora
-----------------------------                    ----------------------------
Secretary


[SEAL]

ATTEST:                                      RICHMOND COUNTY FINANCIAL CORP.


/s/ Carolynn Orisino                         BY: /s/ Michael F. Manzulli
------------------------                         ---------------------------
Secretary

[SEAL]

ATTEST:                                      RICHMOND COUNTY SAVINGS BANK


/s/ Carolynn Orisino                         By: /s/ Michael F. Manzulli
-----------------------------                    ---------------------------
Secretary

[SEAL]

WITNESS:                                     EXECUTIVE

/s/ Carolynn Orisino                         /s/ Anthony E. Burke
------------------------------               ------------------------------
                                             Anthony E. Burke