I submitted my manuscript to a journal one month ago and the status is "with editor" up to now. As the editor of my paper is one of the leading scientists in the field of my research paper (mathematics), I would like to know that can an editor do the review of paper by him/herself without sending to other reviewers?
Yes, an editor can review a manuscript themselves, but you cannot infer an editor is reviewing themselves when a manuscript has been with "a journal [for] one month...and the status is 'with editor' up to now."
The editor may simply not have had the time to look at your manuscript and find a suitable reviewer. Or he may have contacted one or more potential reviewers, but they may not have accepted yet, in which case the editorial system may still show the manuscript as "with editor".
Also, of course, editors do look at manuscripts themselves. They typically don't write full reviews themselves, but they may desk reject, i.e., reject it without even sending it out for review, if the manuscript is obviously not within the journal's scope or of too low quality. ("Obviously" is obviously subjective.)
If you fear a conflict of interest, it would be good to account for this before submission, by either asking the manuscript to be handled by a different editor (assuming the journal has more than one), or by submitting to a different journal.
I am an editor at a mathematics journal which uses a (IMHO) pretty crappy online editorial system. When a paper is assigned to me, I find it much easier to work with referees "off the grid" rather than using the infernal editorial system. (For example, the system insists that I input in an entire page of details before sending a referee request to someone who most likely will ignore the request anyway.) So I don't think you can infer anything at all from the fact that the online status says "with editor."