If a cisco switch is configure as layer 3 switch using eigrp and its has interface vlan 192.168.34.1 and is advertise under eigrp network 192.168.34.0 0.0.0.255. And it's has port configure just as switchport trunk and then I have another switch comfigure as switchport trunk. The question now is I just want to manage that switch so then I configure an svi with IP address 192.168.34.7 255.255.255.0. Will I able to manage that switch with that svi and no ip default gateway command because the layer 3 switch will take care of advertise that 192.168.34.0 network?
Are you asking whether by running eigrp on a layer 3 switch and by enabling eigrp for a switch svi, will you be able to manage without configuring a default gateway on the switch? The answer is yes, provided you have also enabled 'ip routing' on the switch.
Verify using - show ip route, and make sure you have a route back to the ip address of the device you are trying to manage the switch from.
If there is nothing in the routing table - troubleshoot eigrp as you would do for a router.
There are some things in your question that I do not understand well. So let me try to answer what I think I understand and if it does not fully meet your needs then perhaps you can offer some clarification. You have a layer 3 switch and have configured EIGRP to run on one of its VLANs and to advertise that VLAN. You have another switch configured with a trunk connection to the layer 3 switch. You have configured an SVI in the same subnet on the second switch. You have not told us but I am assuming that the SVI on the second switch is in the same VLAN as the one on the layer 3 switch.
You have not told us whether the second switch is also participating in EIGRP and that is an important aspect. If the second switch is running EIGRP for this subnet then you should be able to manage this switch from anywhere and not need a default gateway statement. If the second switch does not run EIGRP and does not have a default gateway statement then you would only be able to manage this switch from other devices in the 192.168.34.0 subnet. Devices in other networks would know how to reach the second switch because EIGRP is advertising the subnet. But the second switch would not know how to reach anything outside of the subnet (without EIGRP and without a default gateway).
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