EXHIBIT 4.10
 
                    AGREEMENT AS TO EXPENSES AND LIABILITIES

     AGREEMENT AS TO EXPENSES AND LIABILITIES (this "Agreement"), dated as of
March 31, 1997, between Greater Bay Bancorp, a California corporation (the
"Company"), and GBB Capital I, a Delaware business trust (the "Trust").

     WHEREAS, the Trust intends to issue its Common Securities (the "Common
Securities") to and receive Junior Subordinated Deferrable Interest Debentures
(the "Debentures") from the Company and to issue and sell 9.75% Cumulative Trust
Preferred Securities (the "Trust Preferred Securities") with such powers,
preferences and special rights and restrictions as are set forth in the Amended
and Restated Trust Agreement of the Trust dated as of March 31, 1997 as the same
may be amended from time to time (the "Trust Agreement");

     WHEREAS, the Company will directly or indirectly own all of the Common
Securities of the Trust and will issue the Debentures;

     NOW, THEREFORE, in consideration of the purchase by each holder of the
Trust Preferred Securities, which purchase the Company hereby agrees shall
benefit the Company and which purchase the Company acknowledges will be made in
reliance upon the execution and delivery of this Agreement, the Company and the
Trust hereby agree as follows:


                                   ARTICLE I

     Section 1.1. Guarantee by the Company.

     Subject to the terms and conditions hereof, the Company hereby irrevocably
and unconditionally guarantees to each person or entity to whom the Trust is now
or hereafter becomes indebted or liable (the "Beneficiaries") the full payment,
when and as due, of any and all Obligations (as hereinafter defined) to such
Beneficiaries. As used herein, "Obligations" means any costs, expenses or
liabilities of the Trust, other than obligations of the Trust to pay to holders
of any Trust Preferred Securities or other similar interests in the Trust the
amounts due such holders pursuant to the terms of the Trust Preferred Securities
or such other similar interests, as the case may be. This Agreement is intended
to be for the benefit of, and to be enforceable by, all such Beneficiaries,
whether or not such Beneficiaries have received notice hereof.

     Section 1.2. Term of Agreement.

     This Agreement shall terminate and be of no further force and effect upon
the later of (a) the date on which full payment has been made of all amounts
payable to all holders of all the Trust Preferred Securities (whether upon
redemption, liquidation, exchange or otherwise) and (b) the date on which there
are no Beneficiaries remaining; provided, however, that this Agreement shall
continue to be effective or shall be reinstated, as the case may be, if at any
time any holder of Trust Preferred Securities or any Beneficiary must restore
payment of any sums paid under the Trust Preferred Securities, under any
Obligation, under the Guarantee Agreement dated the date hereof by the 

 
Company and Wilmington Trust Company, a Delaware banking corporation, as
guarantee trustee or under this Agreement for any reason whatsoever. This
Agreement is continuing, irrevocable, unconditional and absolute.

     Section 1.3. Waiver of Notice.

     The Company hereby waives notice of acceptance of this Agreement and of any
Obligation to which it applies or may apply, and the Company hereby waives
presentment, demand for payment, protest, notice of nonpayment, notice of
dishonor, notice of redemption and all other notices and demands.

     Section 1.4. No Impairment.

     The obligations, covenants, agreements and duties of the Company under this
Agreement shall in no way be affected or impaired by reason of the happening
from time to time of any of the following:

     (a) the extension of time for the payment by the Trust of all or any
portion of the Obligations or for the performance of any other obligation under,
arising out of, or in connection with, the Obligations;

     (b) any failure, omission, delay or lack of diligence on the part of the
Beneficiaries to enforce, assert or exercise any right, privilege, power or
remedy conferred on the Beneficiaries with respect to the Obligations or any
action on the part of the Trust granting indulgence or extension of any kind; or

     (c) the voluntary or involuntary liquidation, dissolution, sale of any
collateral, receivership, insolvency, bankruptcy, assignment for the benefit of
creditors, reorganization, arrangement, composition or readjustment of debt of,
or other similar proceedings affecting, the Trust or any of the assets of the
Trust.

There shall be no obligation of the Beneficiaries to give notice to, or obtain
the consent of, the Company with respect to the happening of any of the
foregoing.

     Section 1.5. Enforcement.

     A Beneficiary may enforce this Agreement directly against the Company and
the Company waives any right or remedy to require that any action be brought
against the Trust or any other person or entity before proceeding against the
Company.

     Section 1.6. Subrogation.

     The Company shall be subrogated to all (if any) rights of the Trust in
respect of any amounts paid to the Beneficiaries by the Company under this
Agreement; provided, however, that the 

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Company shall not (except to the extent required by mandatory provisions of law)
be entitled to enforce or exercise any rights which it may acquire by way of
subrogation or any indemnity, reimbursement or other agreement, in all cases as
a result of payment under this Agreement, if, at the time of any such payment,
any amounts are due and unpaid under this Agreement.


                                   ARTICLE II

     Section 2.1. Binding Effect.

     All guarantees and agreements contained in this Agreement shall bind the
successors, assigns, receivers, trustees and representatives of the Company and
shall inure to the benefit of the Beneficiaries.

     Section 2.2. Amendment.

     So long as there remains any Beneficiary or any Trust Preferred Securities
are outstanding, this Agreement shall not be modified or amended in any manner
adverse to such Beneficiary or to the holders of the Trust Preferred Securities.

     Section 2.3. Notices.

     Any notice, request or other communication required or permitted to be
given hereunder shall be given in writing by delivering the same against receipt
therefor by facsimile transmission (confirmed by mail), telex or by registered
or certified mail, addressed as follows (and if so given, shall be deemed given
when mailed or upon receipt of an answer-back, if sent by telex):

          GBB Capital I                
          c/o Greater Bay Bancorp      
          2860 West Bayshore Road      
          Palo Alto, California 94303  
          Facsimile No.: (415) 813-8211
          Attention:  Steven C. Smith  
                                       
          Greater Bay Bancorp          
          2860 West Bayshore Road      
          Palo Alto, California 94303  
          Facsimile No.: (415) 813-8211
          Attention:  Steven C. Smith   

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     Section 2.4. Choice of Law.

     THIS AGREEMENT SHALL BE GOVERNED BY AND CONSTRUED AND INTERPRETED IN
ACCORDANCE WITH THE LAWS OF THE STATE OF CALIFORNIA (WITHOUT REGARD TO CONFLICT
OF LAW PRINCIPLES).

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     THIS AGREEMENT is executed as of the day and year first above written.

                                      GREATER BAY BANCORP


                                  By: /s/ Steven C. Smith
                                      -------------------------------------
                                      Steven C. Smith
                                      Executive Vice President, Chief Operating 
                                      Officer and Chief Financial Officer


                                      GBB Capital I


                                  By: /s/ Steven C. Smith
                                      -------------------------------------
                                      Steven C. Smith
                                      Administrative Trustee

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