Frobenius Theorem and a Principle for Critical Point under the Constraint of $C^1$ Submanifold in Banach Spaces

TL;DR

This paper extends the Frobenius Theorem and introduces a principle for identifying critical points under $C^1$ submanifold constraints in Banach spaces, advancing the theoretical framework for constrained analysis.

Contribution

It develops a Frobenius Theorem and a critical point principle specifically for $C^1$ submanifolds in Banach spaces, filling a gap in infinite-dimensional analysis.

Findings

Established Frobenius Theorem for $C^1$ submanifolds in Banach spaces

Formulated a principle for critical points under submanifold constraints

Provided theoretical tools for constrained optimization in infinite-dimensional spaces

Abstract

In this paper, we establish the Frobenius Theorem and a Principle for Critical Point under the Constraint of $C^1$ Submanifold in Banach Spaces.