This article concerned on the study of signature submanifolds for curves under Lie group actions SE(2), SA(2) and for surfaces under SE(3). Signature submanifold is a regular submanifold which its coordinate components are differential invariants of an associated manifold under Lie group action, and therefore signature submanifold is a key for solving equivalence problems.

A symplectic Lie group is a with left-invariant form. Its algebra structure that of quasi-Frobenius algebra. In this note, we identify the groupoid analogue group. We call aforementioned \textit{$t$-symplectic groupoid}; $t$ motivated by fact each target fiber $t$-symplectic manifold. For $\mathcal{G}\rightrightarrows M$, show there one-to-one correspondence between algebroid structures on $A\m...

The main target of this article is to study about Lie Groups, Algebras. This will enrich our knowledge Algebraic properties manifolds, how Groups and Algebras are working with their properties. Finally, we have discussed an example by showing all the Algebra,Lie for a special Group Theorem has established. Dhaka Univ. J. Sci. 71(1): 82-86, 2023 (Jan)

Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see §1). In general, this topological structure of A(G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an i...