MathDB

Regular submanifold usually has no Lie group structure

Source: Alibaba Global Math Competition 2021, Problem 13

July 4, 2021

group theorysubmanifoldManifoldgeometrytopologycollege contests

Problem Statement

Let

M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}

, where

n \ge 2

, and

u \cdot v

is the Euclidean inner product of

u

and

v

. Suppose that the topology of

M_n

is induces from

S^n \times S^n

.
(1) Prove that

M_n

is a connected regular submanifold of

S^n \times S^n

.
(2)

M_n

is Lie Group if and only if

n=2