I have been learning about multivariate calculus on my own. I have learned, notation-wise the properties of concave and quasi-concave functions. But I am finding it very difficult to find geometric intuitions for many of the properties.
For example, when reading a specific example, regarding a real valued function $u(x): R^n_+ \rightarrow R$ having the Hessian matrix $H(x)$ that is twice differentiable, it said:
$$y^T H(x) y \leq 0$$
For all vectors $y$ such that
$$\nabla u(x) \cdot y =0$$
The explanation goes, that if we move in a direction tangent to the level curve of the function (because $\nabla u(x) \cdot y =0$, this part I understand), then the value of $u(x)$ will fall (as $y^T H(x) y \leq 0$) -- this second part I know must be true due to logic, but I am unable to develop a geometric intuition in my mind.
Even beyond this example, I find it difficult to relate Hessian matrices with curvature geometrically in my mind.
How can I develop some intuition regarding this. An explanation or referring to any material (book/ notes/ video etc.) that can help me form such intuition would help.