# Question regarding description of a $\lambda$-handle in Matsumoto's “ An Introduction to Morse Theory”

On Page 76 in Y. Matsumoto's "An Introduction to Morse Theory" he introduces an $$m$$-dimensional $$\lambda$$-handle including the following figure: he then says

The lightly shaded area corresponds to the inequalities

$$x_1^2+\cdots+ x_\lambda^2-x_{\lambda+1}^2-\cdots - x_m^2 \le \varepsilon$$ $$x_{\lambda+1}^2+\cdots+ x_m^2\le \delta$$

Remark: the lightly shaded area is supposed to be the "bridge" between the black areas.

My question is: shouldn't the first inequality rather be

$$x_1^2+\cdots+ x_\lambda^2 \le \varepsilon$$

such that the slightly shaded area is described by the inequalities

$$x_1^2+\cdots+ x_\lambda^2 \le \varepsilon$$ $$x_{\lambda+1}^2+\cdots+ x_m^2\le \delta$$

Thanks for any consideration!

I don't have Matsumoto's book in front of me, but I do have Milnor's Morse Theory, whose figures 6 and 7 (pp. 18-19) seem to describe the same thing. The general topic is how $$M^a$$ is obtained from $$M^0$$, where $$M^t=\{x\in M: f(x)\le t\}$$ is the sub-level set (my made-up term; Milnor does't seem to provide one) for some $$a>0$$. The union of the two black regions represents $$M^0$$ and their union with the lightly shaded region represents $$M^a$$.
Pictorially, the shaded region is close to the horizontal "axis", deformable to a tube connecting the two black regions. The inequality not in dispute is $$x_{\lambda+1}^2+\cdots x_m^2\le\delta$$, which forces the shaded region to be a subset of the tube around the horizontal axis. M's inequality $$x_1^2+\cdots x_\lambda^2-x_{\lambda+1}^2\cdots -x_m^2 \le \epsilon$$ says $$f(x)\le \epsilon$$, that is, cuts out $$M^\epsilon$$.
The OP's interpretation that the constraint should be $$x_{\lambda+1}^2+\cdots +x_m^2 \le \epsilon$$ would restrict the shaded region to a tube about the "vertical" axis, as well, so the shaded region would be the the intersections of the vertical tube and the horizontal tube, a small blob about the origin, not connecting the two black regions.
I'm sure the OP understands the general thrust of the "lowering sea levels creates a land bridge or causeway (the shaded region) between Asia and America (the black regions)" story given in popular expositions of Morse theory. The $$\epsilon$$ inequality says that in this picture the causeway connects the two continents; the $$\delta$$ inequality says that the connection is narrow.